What is the general solution for solvefor b,a=cos(-1(a/(sqrt(a^2+b^2)))) ?

The general solution for solvefor b,a=cos(-1(a/(sqrt(a^2+b^2)))) is

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solvefor b,a=cos(-1(a/(sqrt(a^2+b^2))))

Solution

solve for

b, a = cos (- 1 (\frac{a}{a ^{2} + b ^{2}}))

Solution

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Solution steps

a = cos (- 1 \cdot (\frac{a}{a ^{2} + b ^{2}}))

Switch sides

cos (- 1 \cdot \frac{a}{a ^{2} + b ^{2}}) = a

Apply trig inverse properties

cos (- 1 \cdot \frac{a}{a ^{2} + b ^{2}}) = a

General solutions for

cos (- 1 \frac{a}{a ^{2} + b ^{2}}) = a

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn, - 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn, - 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

Solve

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn

Expand

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} : - \frac{a}{a ^{2} + b ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}}

Multiply:

1 \cdot \frac{a}{a ^{2} + b ^{2}} = \frac{a}{a ^{2} + b ^{2}}

= - \frac{a}{a ^{2} + b ^{2}}

- \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn

Square both sides:

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

- \frac{a}{a ^{2} + b ^{2}} = arccos (a) + 2 πn

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (arccos (a) + 2 πn)^{2}

Expand

(- \frac{a}{a ^{2} + b ^{2}})^{2} : \frac{a ^{2}}{a ^{2} + b ^{2}}

(- \frac{a}{a ^{2} + b ^{2}})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (\frac{a}{a ^{2} + b ^{2}})^{2}

= (\frac{a}{a ^{2} + b ^{2}})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{a ^{2}}{( a ^{2} + b ^{2} ) ^{2}}

(a^{2} + b^{2})^{2} : a^{2} + b^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((a^{2} + b^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (a^{2} + b^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= a^{2} + b^{2}

= \frac{a ^{2}}{a ^{2} + b ^{2}}

Expand

(arccos (a) + 2 πn)^{2} : arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

(arccos (a) + 2 πn)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = arccos (a), b = 2 πn

= arccos^{2} (a) + 2 arccos (a) \cdot 2 πn + (2 πn)^{2}

Simplify

arccos^{2} (a) + 2 arccos (a) \cdot 2 πn + (2 πn)^{2} : arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

arccos^{2} (a) + 2 arccos (a) \cdot 2 πn + (2 πn)^{2}

2 arccos (a) \cdot 2 πn = 4 πn arccos (a)

2 arccos (a) \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn arccos (a)

(2 πn)^{2} = 4 π^{2} n^{2}

(2 πn)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} π^{2} n^{2}

2^{2} = 4

= 4 π^{2} n^{2}

= arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

= arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

Solve

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2} : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

Multiply both sides by

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

Multiply both sides by

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} (a^{2} + b^{2}) = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Simplify

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Solve

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Switch sides

arccos^{2} (a) (a^{2} + b^{2}) + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expand

arccos^{2} (a) (a^{2} + b^{2}) : a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

arccos^{2} (a) (a^{2} + b^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = arccos^{2} (a), b = a^{2}, c = b^{2}

= arccos^{2} (a) a^{2} + arccos^{2} (a) b^{2}

= a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expand

4 πn arccos (a) (a^{2} + b^{2}) : 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2}

4 πn arccos (a) (a^{2} + b^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 4 πn arccos (a), b = a^{2}, c = b^{2}

= 4 πn arccos (a) a^{2} + 4 πn arccos (a) b^{2}

= 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expand

4 π^{2} n^{2} (a^{2} + b^{2}) : 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

4 π^{2} n^{2} (a^{2} + b^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 4 π^{2} n^{2}, b = a^{2}, c = b^{2}

= 4 π^{2} n^{2} a^{2} + 4 π^{2} n^{2} b^{2}

= 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Move

a^{2} arccos^{2} (a)

to the right side

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Subtract

a^{2} arccos^{2} (a)

from both sides

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - a^{2} arccos^{2} (a) = a^{2} - a^{2} arccos^{2} (a)

Simplify

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Move

4 π a^{2} n arccos (a)

to the right side

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Subtract

4 π a^{2} n arccos (a)

from both sides

arccos^{2} (a) b^{2} + 4 π a^{2} n arccos (a) + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - 4 π a^{2} n arccos (a) = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

Simplify

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

Move

4 π^{2} a^{2} n^{2}

to the right side

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a)

Subtract

4 π^{2} a^{2} n^{2}

from both sides

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - 4 π^{2} a^{2} n^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Simplify

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Factor

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} : b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2})

arccos^{2} (a) b^{2} + 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2}

Factor out common term

b^{2}

= b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2})

b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Divide both sides by

arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

b^{2} (arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) - 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Divide both sides by

arccos^{2} (a) + 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{b ^{2} ( arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2} )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplify

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplify

\frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combine the fractions

\frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 πn a ^{2} arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 π a ^{2} n arccos ( a ) - 4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplify

- \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- \frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combine the fractions

\frac{a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 πn a ^{2} arccos ( a )}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Solve

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

Expand

- 1 \cdot \frac{a}{a ^{2} + b ^{2}} : - \frac{a}{a ^{2} + b ^{2}}

- 1 \cdot \frac{a}{a ^{2} + b ^{2}}

Multiply:

1 \cdot \frac{a}{a ^{2} + b ^{2}} = \frac{a}{a ^{2} + b ^{2}}

= - \frac{a}{a ^{2} + b ^{2}}

- \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

Square both sides:

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

- \frac{a}{a ^{2} + b ^{2}} = - arccos (a) + 2 πn

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (- arccos (a) + 2 πn)^{2}

Expand

(- \frac{a}{a ^{2} + b ^{2}})^{2} : \frac{a ^{2}}{a ^{2} + b ^{2}}

(- \frac{a}{a ^{2} + b ^{2}})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{a}{a ^{2} + b ^{2}})^{2} = (\frac{a}{a ^{2} + b ^{2}})^{2}

= (\frac{a}{a ^{2} + b ^{2}})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{a ^{2}}{( a ^{2} + b ^{2} ) ^{2}}

(a^{2} + b^{2})^{2} : a^{2} + b^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((a^{2} + b^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (a^{2} + b^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= a^{2} + b^{2}

= \frac{a ^{2}}{a ^{2} + b ^{2}}

Expand

(- arccos (a) + 2 πn)^{2} : arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

(- arccos (a) + 2 πn)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - arccos (a), b = 2 πn

= (- arccos (a))^{2} + 2 (- arccos (a)) \cdot 2 πn + (2 πn)^{2}

Simplify

(- arccos (a))^{2} + 2 (- arccos (a)) \cdot 2 πn + (2 πn)^{2} : arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

(- arccos (a))^{2} + 2 (- arccos (a)) \cdot 2 πn + (2 πn)^{2}

Remove parentheses:

(- a) = - a

= (- arccos (a))^{2} - 2 arccos (a) \cdot 2 πn + (2 πn)^{2}

(- arccos (a))^{2} = arccos^{2} (a)

(- arccos (a))^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- arccos (a))^{2} = arccos^{2} (a)

= arccos^{2} (a)

2 arccos (a) \cdot 2 πn = 4 πn arccos (a)

2 arccos (a) \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn arccos (a)

(2 πn)^{2} = 4 π^{2} n^{2}

(2 πn)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} π^{2} n^{2}

2^{2} = 4

= 4 π^{2} n^{2}

= arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

= arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

Solve

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2} : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

Multiply both sides by

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} = arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

Multiply both sides by

a^{2} + b^{2}

\frac{a ^{2}}{a ^{2} + b ^{2}} (a^{2} + b^{2}) = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Simplify

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Solve

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) : b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

a^{2} = arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2})

Switch sides

arccos^{2} (a) (a^{2} + b^{2}) - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expand

arccos^{2} (a) (a^{2} + b^{2}) : a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

arccos^{2} (a) (a^{2} + b^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = arccos^{2} (a), b = a^{2}, c = b^{2}

= arccos^{2} (a) a^{2} + arccos^{2} (a) b^{2}

= a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 πn arccos (a) (a^{2} + b^{2}) + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expand

- 4 πn arccos (a) (a^{2} + b^{2}) : - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2}

- 4 πn arccos (a) (a^{2} + b^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = - 4 πn arccos (a), b = a^{2}, c = b^{2}

= - 4 πn arccos (a) a^{2} + (- 4 πn arccos (a)) b^{2}

Apply minus-plus rules

+ (- a) = - a

= - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} (a^{2} + b^{2}) = a^{2}

Expand

4 π^{2} n^{2} (a^{2} + b^{2}) : 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

4 π^{2} n^{2} (a^{2} + b^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 4 π^{2} n^{2}, b = a^{2}, c = b^{2}

= 4 π^{2} n^{2} a^{2} + 4 π^{2} n^{2} b^{2}

= 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2}

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Move

a^{2} arccos^{2} (a)

to the right side

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2}

Subtract

a^{2} arccos^{2} (a)

from both sides

a^{2} arccos^{2} (a) + arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - a^{2} arccos^{2} (a) = a^{2} - a^{2} arccos^{2} (a)

Simplify

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Move

4 π a^{2} n arccos (a)

to the right side

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a)

Add

4 π a^{2} n arccos (a)

to both sides

arccos^{2} (a) b^{2} - 4 π a^{2} n arccos (a) - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} + 4 π a^{2} n arccos (a) = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

Simplify

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

Move

4 π^{2} a^{2} n^{2}

to the right side

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a)

Subtract

4 π^{2} a^{2} n^{2}

from both sides

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} a^{2} n^{2} + 4 π^{2} n^{2} b^{2} - 4 π^{2} a^{2} n^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Simplify

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Factor

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2} : b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2})

arccos^{2} (a) b^{2} - 4 πn arccos (a) b^{2} + 4 π^{2} n^{2} b^{2}

Factor out common term

b^{2}

= b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2})

b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Divide both sides by

arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

b^{2} (arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}) = a^{2} - a^{2} arccos^{2} (a) + 4 π a^{2} n arccos (a) - 4 π^{2} a^{2} n^{2}

Divide both sides by

arccos^{2} (a) - 4 πn arccos (a) + 4 π^{2} n^{2}

\frac{b ^{2} ( arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2} )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplify

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b^{2} = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplify

\frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

\frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combine the fractions

\frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 πn a ^{2} arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 π a ^{2} n arccos ( a ) - 4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Simplify

- \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

- \frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 π a ^{2} n arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} a ^{2} n ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Combine the fractions

\frac{a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{a ^{2} arccos ^{2} ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} + \frac{4 πn a ^{2} arccos ( a )}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} - \frac{4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}} : \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= - \frac{a ^{2} + 4 πn a ^{2} arccos ( a ) - a ^{2} arccos ^{2} ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

= - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) - 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) + 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}, b = - \frac{a ^{2} - a ^{2} arccos ^{2} ( a ) + 4 πn a ^{2} arccos ( a ) - 4 π ^{2} n ^{2} a ^{2}}{arccos ^{2} ( a ) - 4 πn arccos ( a ) + 4 π ^{2} n ^{2}}

Popular Examples

(1/2 sin(2x))^2= 1/4 sin^2(4x)

(\frac{1}{2} sin (2 x))^{2} = \frac{1}{4} sin^{2} (4 x)

8sin^2(x)-3sin(x)-2=0

8 sin^{2} (x) - 3 sin (x) - 2 = 0

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2 sin^{2} (x) 3 = 7 sin (x), 0 \leq x \leq 2 π

6sin(x)+12=15

6 sin (x) + 12 = 15

cot^2(x)cos(x)=cos(x)

cot^{2} (x) cos (x) = cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for solvefor b,a=cos(-1(a/(sqrt(a^2+b^2)))) ?
The general solution for solvefor b,a=cos(-1(a/(sqrt(a^2+b^2)))) is