What is the general solution for solvefor x,f=arctan(x/(sqrt(1-x^2))) ?

The general solution for solvefor x,f=arctan(x/(sqrt(1-x^2))) is x=(tan(f))/(sqrt(1+tan^2(f)))

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Popular Trigonometry >

solvefor x,f=arctan(x/(sqrt(1-x^2)))

Solution

solvefor

Solution

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

Solution steps

f = arctan (\frac{x}{1 - x ^{2}})

Switch sides

arctan (\frac{x}{1 - x ^{2}}) = f

Apply trig inverse properties

arctan (\frac{x}{1 - x ^{2}}) = f

arctan (x) = a \Rightarrow x = tan (a)

\frac{x}{1 - x ^{2}} = tan (f)

\frac{x}{1 - x ^{2}} = tan (f)

Solve

\frac{x}{1 - x ^{2}} = tan (f) : x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

\frac{x}{1 - x ^{2}} = tan (f)

Multiply both sides by

1 - x^{2}

\frac{x}{1 - x ^{2}} 1 - x^{2} = tan (f) 1 - x^{2}

Simplify

x = tan (f) 1 - x^{2}

Square both sides:

x^{2} = tan^{2} (f) - x^{2} tan^{2} (f)

x = tan (f) 1 - x^{2}

x^{2} = (tan (f) 1 - x^{2})^{2}

Expand

(tan (f) 1 - x^{2})^{2} : tan^{2} (f) - x^{2} tan^{2} (f)

(tan (f) 1 - x^{2})^{2}

Apply exponent rule:

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

= tan^{2} (f) (1 - x^{2})^{2}

(1 - x^{2})^{2} : 1 - x^{2}

Apply radical rule:

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

= ((1 - x^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 - x^{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

= 1 - x^{2}

= tan^{2} (f) (1 - x^{2})

Expand

(1 - x^{2}) tan^{2} (f) : tan^{2} (f) - x^{2} tan^{2} (f)

(1 - x^{2}) tan^{2} (f)

= tan^{2} (f) (1 - x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = tan^{2} (f), b = 1, c = x^{2}

= tan^{2} (f) \cdot 1 - tan^{2} (f) x^{2}

= 1 \cdot tan^{2} (f) - x^{2} tan^{2} (f)

Multiply:

1 \cdot tan^{2} (f) = tan^{2} (f)

= tan^{2} (f) - x^{2} tan^{2} (f)

= tan^{2} (f) - x^{2} tan^{2} (f)

x^{2} = tan^{2} (f) - x^{2} tan^{2} (f)

x^{2} = tan^{2} (f) - x^{2} tan^{2} (f)

Solve

x^{2} = tan^{2} (f) - x^{2} tan^{2} (f) : x = \frac{tan ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

x^{2} = tan^{2} (f) - x^{2} tan^{2} (f)

Move

x^{2} tan^{2} (f)

to the left side

x^{2} = tan^{2} (f) - x^{2} tan^{2} (f)

Add

x^{2} tan^{2} (f)

to both sides

x^{2} + x^{2} tan^{2} (f) = tan^{2} (f) - x^{2} tan^{2} (f) + x^{2} tan^{2} (f)

Simplify

x^{2} + x^{2} tan^{2} (f) = tan^{2} (f)

x^{2} + x^{2} tan^{2} (f) = tan^{2} (f)

Factor

x^{2} + x^{2} tan^{2} (f) : x^{2} (1 + tan^{2} (f))

x^{2} + x^{2} tan^{2} (f)

Factor out common term

x^{2}

= x^{2} (1 + tan^{2} (f))

x^{2} (1 + tan^{2} (f)) = tan^{2} (f)

Divide both sides by

1 + tan^{2} (f)

x^{2} (1 + tan^{2} (f)) = tan^{2} (f)

Divide both sides by

1 + tan^{2} (f)

\frac{x ^{2} ( 1 + tan ^{2} ( f ) )}{1 + tan ^{2} ( f )} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Simplify

x^{2} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

x^{2} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

For

x^{2} = f (a)

the solutions are

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

x = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Simplify

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )} : \frac{tan ( f )}{1 + tan ^{2} ( f )}

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Apply radical rule:

assuming

a \geq 0

tan^{2} (f) = tan (f)

= \frac{tan ( f )}{1 + tan ^{2} ( f )}

Simplify

- \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )} : - \frac{tan ( f )}{1 + tan ^{2} ( f )}

- \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Simplify

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )} : \frac{tan ( f )}{1 + tan ^{2} ( f )}

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Apply radical rule:

assuming

a \geq 0

tan^{2} (f) = tan (f)

= \frac{tan ( f )}{1 + tan ^{2} ( f )}

= - \frac{tan ( f )}{tan ^{2} ( f ) + 1}

= - \frac{tan ( f )}{1 + tan ^{2} ( f )}

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

Verify Solutions:

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

True

, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

False

Check the solutions by plugging them into

\frac{x}{1 - x ^{2}} = tan (f)

Remove the ones that don't agree with the equation.

Plug in

x = \frac{tan ( f )}{1 + tan ^{2} ( f )} :

True

\frac{( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} )}{1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f)

Simplify

\frac{( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} )}{1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} : tan (f)

\frac{\frac{t a n ( f )}{1 + t a n ^{2} ( f )}}{1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}}

Apply the fraction rule:

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

= \frac{tan ( f )}{1 + tan ^{2} ( f ) 1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}}

1 - (\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2} = \frac{1}{1 + tan ^{2} ( f )}

1 - (\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2}

(\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

(\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2}

Apply exponent rule:

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

= \frac{tan ^{2} ( f )}{( 1 + tan ^{2} ( f ) ) ^{2}}

(1 + tan^{2} (f))^{2} : 1 + tan^{2} (f)

Apply radical rule:

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

= ((1 + tan^{2} (f))^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 + tan^{2} (f))^{\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

= 1 + tan^{2} (f)

= \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

= 1 - \frac{tan ^{2} ( f )}{tan ^{2} ( f ) + 1}

Join

1 - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )} : \frac{1}{1 + tan ^{2} ( f )}

1 - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Convert element to fraction:

1 = \frac{1 ( 1 + tan ^{2} ( f ) )}{1 + tan ^{2} ( f )}

= \frac{1 \cdot ( 1 + tan ^{2} ( f ) )}{1 + tan ^{2} ( f )} - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot ( 1 + tan ^{2} ( f ) ) - tan ^{2} ( f )}{1 + tan ^{2} ( f )}

1 \cdot (1 + tan^{2} (f)) - tan^{2} (f) = 1

1 \cdot (1 + tan^{2} (f)) - tan^{2} (f)

1 \cdot (1 + tan^{2} (f)) = 1 + tan^{2} (f)

1 \cdot (1 + tan^{2} (f))

Multiply:

1 \cdot (1 + tan^{2} (f)) = (1 + tan^{2} (f))

= 1 + tan^{2} (f)

Remove parentheses:

(a) = a

= 1 + tan^{2} (f)

= 1 + tan^{2} (f) - tan^{2} (f)

Add similar elements:

tan^{2} (f) - tan^{2} (f) = 0

= 1

= \frac{1}{1 + tan ^{2} ( f )}

= \frac{1}{1 + tan ^{2} ( f )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{1 + tan ^{2} ( f )}

Apply rule

1 = 1

= \frac{1}{1 + tan ^{2} ( f )}

= \frac{tan ( f )}{\frac{1}{t a n ^{2} ( f ) + 1} tan ^{2} ( f ) + 1}

Multiply

1 + tan^{2} (f) \frac{1}{1 + tan ^{2} ( f )} : 1

1 + tan^{2} (f) \frac{1}{1 + tan ^{2} ( f )}

Multiply fractions:

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

= \frac{1 \cdot 1 + tan ^{2} ( f )}{1 + tan ^{2} ( f )}

Cancel the common factor:

1 + tan^{2} (f)

= 1

= \frac{tan ( f )}{1}

Apply rule

\frac{a}{1} = a

= tan (f)

tan (f) = tan (f)

True

Plug

x = - \frac{tan ( f )}{1 + tan ^{2} ( f )} : \frac{- \frac{t a n ( f )}{1 + t a n ^{2} ( f )}}{1 - ( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f) \Rightarrow

False

\frac{( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} )}{1 - ( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f)

Solve by substitution

\frac{- \frac{t a n ( f )}{1 + t a n ^{2} ( f )}}{1 - ( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f)

Let:

tan (f) = u

\frac{- \frac{u}{1 + u ^{2}}}{1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}} = u

\frac{- \frac{u}{1 + u ^{2}}}{1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}} = u :

True for all

u

\frac{- \frac{u}{1 + u ^{2}}}{1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}} = u

Multiply both sides by

1 - (\frac{u}{1 + u ^{2}})^{2}

\frac{- \frac{u}{1 + u ^{2}}}{1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}} 1 - (\frac{u}{1 + u ^{2}})^{2} = u 1 - (\frac{u}{1 + u ^{2}})^{2}

Simplify

- \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}} = u 1 - (\frac{u}{1 + u ^{2}})^{2}

Square both sides:

\frac{u ^{2}}{1 + u ^{2}} = \frac{u ^{2}}{1 + u ^{2}}

- \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}} = u 1 - (\frac{u}{1 + u ^{2}})^{2}

- \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}}^{2} = u 1 - (\frac{u}{1 + u ^{2}})^{2}^{2}

Expand

- \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}}^{2} : \frac{u ^{2}}{1 + u ^{2}}

- \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}}^{2}

Apply exponent rule:

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

n

is even

- \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}}^{2} = \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}}^{2}

= \frac{1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u}{1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2}}^{2}

Apply exponent rule:

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

= \frac{( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u ) ^{2}}{( 1 + u ^{2} 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} ) ^{2}}

Apply exponent rule:

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

1 + u^{2} 1 - (- \frac{u}{1 + u ^{2}})^{2}^{2} = (1 + u^{2})^{2} 1 - (- \frac{u}{1 + u ^{2}})^{2}^{2}

= \frac{( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} u ) ^{2}}{( 1 + u ^{2} ) ^{2} ( 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} ) ^{2}}

Apply exponent rule:

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

1 - (\frac{u}{1 + u ^{2}})^{2} u^{2} = u^{2} 1 - (\frac{u}{1 + u ^{2}})^{2}^{2}

= \frac{u ^{2} ( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} ) ^{2}}{( 1 + u ^{2} ) ^{2} ( 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} ) ^{2}}

(1 + u^{2})^{2} : 1 + u^{2}

Apply radical rule:

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

= ((1 + u^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 + u^{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

= 1 + u^{2}

= \frac{( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} ) ^{2} u ^{2}}{( 1 + u ^{2} ) ( 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} ) ^{2}}

1 - (\frac{u}{1 + u ^{2}})^{2}^{2} : 1 - (\frac{u}{1 + u ^{2}})^{2}

Apply radical rule:

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

= (1 - (\frac{u}{1 + u ^{2}})^{2})^{\frac{1}{2}}^{2}

Apply exponent rule:

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

= (1 - (\frac{u}{1 + u ^{2}})^{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

= 1 - (\frac{u}{1 + u ^{2}})^{2}

= \frac{( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} ) u ^{2}}{( 1 + u ^{2} ) ( 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} ) ^{2}}

1 - (- \frac{u}{1 + u ^{2}})^{2}^{2} : 1 - (- \frac{u}{1 + u ^{2}})^{2}

Apply radical rule:

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

= (1 - (- \frac{u}{1 + u ^{2}})^{2})^{\frac{1}{2}}^{2}

Apply exponent rule:

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

= (1 - (- \frac{u}{1 + u ^{2}})^{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

= 1 - (- \frac{u}{1 + u ^{2}})^{2}

= \frac{( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} ) u ^{2}}{( 1 + u ^{2} ) ( 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} )}

Expand

\frac{( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} ) u ^{2}}{( 1 + u ^{2} ) ( 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} )} : \frac{u ^{2}}{1 + u ^{2}}

\frac{( 1 - ( \frac{u}{1 + u ^{2}} ) ^{2} ) u ^{2}}{( 1 + u ^{2} ) ( 1 - ( - \frac{u}{1 + u ^{2}} ) ^{2} )}

(1 + u^{2}) (1 - (- \frac{u}{1 + u ^{2}})^{2}) = (1 + u^{2}) (1 - (\frac{u}{1 + u ^{2}})^{2})

(1 + u^{2}) (1 - (- \frac{u}{1 + u ^{2}})^{2})

Apply exponent rule:

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

n

is even

(- \frac{u}{u ^{2} + 1})^{2} = (\frac{u}{1 + u ^{2}})^{2}

= (u^{2} + 1) (- (\frac{u}{u ^{2} + 1})^{2} + 1)

= \frac{u ^{2} ( - ( \frac{u}{u ^{2} + 1} ) ^{2} + 1 )}{( u ^{2} + 1 ) ( - ( \frac{u}{u ^{2} + 1} ) ^{2} + 1 )}

Cancel the common factor:

1 - (\frac{u}{1 + u ^{2}})^{2}

= \frac{u ^{2}}{1 + u ^{2}}

= \frac{u ^{2}}{1 + u ^{2}}

Expand

u 1 - (\frac{u}{1 + u ^{2}})^{2}^{2} : \frac{u ^{2}}{1 + u ^{2}}

u 1 - (\frac{u}{1 + u ^{2}})^{2}^{2}

Apply exponent rule:

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

= u^{2} 1 - (\frac{u}{1 + u ^{2}})^{2}^{2}

1 - (\frac{u}{1 + u ^{2}})^{2}^{2} : 1 - (\frac{u}{1 + u ^{2}})^{2}

Apply radical rule:

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

= (1 - (\frac{u}{1 + u ^{2}})^{2})^{\frac{1}{2}}^{2}

Apply exponent rule:

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

= (1 - (\frac{u}{1 + u ^{2}})^{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

= 1 - (\frac{u}{1 + u ^{2}})^{2}

= u^{2} (1 - (\frac{u}{1 + u ^{2}})^{2})

Expand

(1 - (\frac{u}{1 + u ^{2}})^{2}) u^{2} : \frac{u ^{2}}{1 + u ^{2}}

(1 - (\frac{u}{1 + u ^{2}})^{2}) u^{2}

(\frac{u}{1 + u ^{2}})^{2} = \frac{u ^{2}}{1 + u ^{2}}

(\frac{u}{1 + u ^{2}})^{2}

Apply exponent rule:

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

= \frac{u ^{2}}{( 1 + u ^{2} ) ^{2}}

(1 + u^{2})^{2} : 1 + u^{2}

Apply radical rule:

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

= ((1 + u^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 + u^{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

= 1 + u^{2}

= \frac{u ^{2}}{1 + u ^{2}}

= u^{2} (- \frac{u ^{2}}{u ^{2} + 1} + 1)

= u^{2} (1 - \frac{u ^{2}}{1 + u ^{2}})

Apply the distributive law:

a (b - c) = ab - a c

a = u^{2}, b = 1, c = \frac{u ^{2}}{1 + u ^{2}}

= u^{2} \cdot 1 - u^{2} \frac{u ^{2}}{1 + u ^{2}}

= 1 \cdot u^{2} - \frac{u ^{2}}{1 + u ^{2}} u^{2}

1 \cdot u^{2} = u^{2}

1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= u^{2}

\frac{u ^{2}}{1 + u ^{2}} u^{2} = \frac{u ^{4}}{1 + u ^{2}}

\frac{u ^{2}}{1 + u ^{2}} u^{2}

Multiply fractions:

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

= \frac{u ^{2} u ^{2}}{1 + u ^{2}}

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

= \frac{u ^{4}}{1 + u ^{2}}

= u^{2} - \frac{u ^{4}}{u ^{2} + 1}

Convert element to fraction:

u^{2} = \frac{u ^{2} ( 1 + u ^{2} )}{1 + u ^{2}}

= - \frac{u ^{4}}{1 + u ^{2}} + \frac{u ^{2} ( 1 + u ^{2} )}{1 + u ^{2}}

Since the denominators are equal, combine the fractions:

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

= \frac{- u ^{4} + u ^{2} ( 1 + u ^{2} )}{1 + u ^{2}}

Expand

- u^{4} + u^{2} (1 + u^{2}) : u^{2}

- u^{4} + u^{2} (1 + u^{2})

Expand

u^{2} (1 + u^{2}) : u^{2} + u^{4}

u^{2} (1 + u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = u^{2}, b = 1, c = u^{2}

= u^{2} \cdot 1 + u^{2} u^{2}

= 1 \cdot u^{2} + u^{2} u^{2}

Simplify

1 \cdot u^{2} + u^{2} u^{2} : u^{2} + u^{4}

1 \cdot u^{2} + u^{2} u^{2}

1 \cdot u^{2} = u^{2}

1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= u^{2}

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

= u^{2} + u^{4}

= u^{2} + u^{4}

= - u^{4} + u^{2} + u^{4}

Simplify

- u^{4} + u^{2} + u^{4} : u^{2}

- u^{4} + u^{2} + u^{4}

Group like terms

= - u^{4} + u^{4} + u^{2}

Add similar elements:

- u^{4} + u^{4} = 0

= u^{2}

= u^{2}

= \frac{u ^{2}}{1 + u ^{2}}

= \frac{u ^{2}}{1 + u ^{2}}

\frac{u ^{2}}{1 + u ^{2}} = \frac{u ^{2}}{1 + u ^{2}}

\frac{u ^{2}}{1 + u ^{2}} = \frac{u ^{2}}{1 + u ^{2}}

Solve

\frac{u ^{2}}{1 + u ^{2}} = \frac{u ^{2}}{1 + u ^{2}} :

True for all

u

\frac{u ^{2}}{1 + u ^{2}} = \frac{u ^{2}}{1 + u ^{2}}

Subtract

\frac{u ^{2}}{1 + u ^{2}}

from both sides

\frac{u ^{2}}{1 + u ^{2}} - \frac{u ^{2}}{1 + u ^{2}} = \frac{u ^{2}}{1 + u ^{2}} - \frac{u ^{2}}{1 + u ^{2}}

Simplify

0 = 0

Both sides are equal

True for all u

True for all u

Substitute back

u = tan (f)

True for all tan (f)

True for all tan (f)

tan (f) =

True for all

u \in R : f = arctan (True for all u \in R) + πn

tan (f) = True for all u \in R

Apply trig inverse properties

tan (f) = True for all u \in R

General solutions for

tan (f) =

True for all

u \in R

tan (x) = a \Rightarrow x = arctan (a) + πn

f = arctan (True for all u \in R) + πn

f = arctan (True for all u \in R) + πn

Combine all the solutions

f = arctan (True for all u \in R) + πn

Since the equation is undefined for:

arctan (True for all u \in R) + πn

No Solution for f \in R

The solution is

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,f=arctan(x/(sqrt(1-x^2))) ?
The general solution for solvefor x,f=arctan(x/(sqrt(1-x^2))) is x=(tan(f))/(sqrt(1+tan^2(f)))