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tan^2(x)+csc^2(x)=3

Solution

tan^{2} (x) + csc^{2} (x) = 3

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

tan^{2} (x) + csc^{2} (x) = 3

Subtract

3

from both sides

tan^{2} (x) + csc^{2} (x) - 3 = 0

Rewrite using trig identities

- 3 + csc^{2} (x) + tan^{2} (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

= - 3 + csc^{2} (x) + (\frac{1}{cot ( x )})^{2}

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

(\frac{1}{cot ( x )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= - 3 + csc^{2} (x) + \frac{1}{cot ^{2} ( x )}

Use the Pythagorean identity:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= - 3 + csc^{2} (x) + \frac{1}{csc ^{2} ( x ) - 1}

- 3 + csc^{2} (x) + \frac{1}{- 1 + csc ^{2} ( x )} = 0

Solve by substitution

- 3 + csc^{2} (x) + \frac{1}{- 1 + csc ^{2} ( x )} = 0

Let:

csc (x) = u

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0 : u = 2, u = - 2

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0

Multiply both sides by

- 1 + u^{2}

- 3 + u^{2} + \frac{1}{- 1 + u ^{2}} = 0

Multiply both sides by

- 1 + u^{2}

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + \frac{1}{- 1 + u ^{2}} (- 1 + u^{2}) = 0 \cdot (- 1 + u^{2})

Simplify

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + \frac{1}{- 1 + u ^{2}} (- 1 + u^{2}) = 0 \cdot (- 1 + u^{2})

Simplify

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

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

Multiply fractions:

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

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

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

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

Multiply:

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

= (- 1 + u^{2})

Remove parentheses:

(- a) = - a

= - 1 + u^{2}

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

Apply rule

\frac{a}{a} = 1

= 1

Simplify

0 \cdot (- 1 + u^{2}) : 0

0 \cdot (- 1 + u^{2})

Apply rule

0 \cdot a = 0

= 0

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

Solve

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0 : u = 2, u = - 2

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 = 0

Expand

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1 : u^{4} - 4 u^{2} + 4

- 3 (- 1 + u^{2}) + u^{2} (- 1 + u^{2}) + 1

Expand

- 3 (- 1 + u^{2}) : 3 - 3 u^{2}

- 3 (- 1 + u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = - 3, b = - 1, c = u^{2}

= - 3 (- 1) + (- 3) u^{2}

Apply minus-plus rules

- (- a) = a, + (- a) = - a

= 3 \cdot 1 - 3 u^{2}

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 u^{2}

= 3 - 3 u^{2} + u^{2} (- 1 + u^{2}) + 1

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

Apply minus-plus rules

+ (- a) = - a

= - 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}

= 3 - 3 u^{2} - u^{2} + u^{4} + 1

Simplify

3 - 3 u^{2} - u^{2} + u^{4} + 1 : u^{4} - 4 u^{2} + 4

3 - 3 u^{2} - u^{2} + u^{4} + 1

Add similar elements:

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

= 3 - 4 u^{2} + u^{4} + 1

Group like terms

= u^{4} - 4 u^{2} + 3 + 1

Add the numbers:

3 + 1 = 4

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

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

u^{4} - 4 u^{2} + 4 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} - 4 v + 4 = 0

Solve

v^{2} - 4 v + 4 = 0 : v = 2

v^{2} - 4 v + 4 = 0

Solve with the quadratic formula

v^{2} - 4 v + 4 = 0

Quadratic Equation Formula:

For

a = 1, b = - 4, c = 4

v_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

(- 4)^{2} - 4 \cdot 1 \cdot 4 = 0

(- 4)^{2} - 4 \cdot 1 \cdot 4

Apply exponent rule:

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

n

is even

(- 4)^{2} = 4^{2}

= 4^{2} - 4 \cdot 1 \cdot 4

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

v_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 \cdot 1}

v = \frac{- ( - 4 )}{2 \cdot 1}

\frac{- ( - 4 )}{2 \cdot 1} = 2

\frac{- ( - 4 )}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

v = 2

The solution to the quadratic equation is:

v = 2

v = 2

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 2 : u = 2, u = - 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

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

u = 2, u = - 2

The solutions are

u = 2, u = - 2

u = 2, u = - 2

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

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

and compare to zero

Solve

- 1 + u^{2} = 0 : u = 1, u = - 1

- 1 + u^{2} = 0

Move

1

to the right side

- 1 + u^{2} = 0

Add

1

to both sides

- 1 + u^{2} + 1 = 0 + 1

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u = 2, u = - 2

Substitute back

u = csc (x)

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

csc (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = 2

General solutions for

csc (x) = 2

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = - 2 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

csc (x) = - 2

General solutions for

csc (x) = - 2

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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