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(sec^2(θ)-1)/(1-cos^2(θ))-1=cot^2(θ)

Solution

\frac{sec ^{2} ( θ ) - 1}{1 - cos ^{2} ( θ )} - 1 = cot^{2} (θ)

Solution

θ = 0.78539 \dots + 2 πn, θ = 2 π - 0.78539 \dots + 2 πn, θ = 2.35619 \dots + 2 πn, θ = - 2.35619 \dots + 2 πn

Degrees

θ = 4 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n, θ = 13 5^{\circ} + 36 0^{\circ} n, θ = - 13 5^{\circ} + 36 0^{\circ} n

Solution steps

\frac{sec ^{2} ( θ ) - 1}{1 - cos ^{2} ( θ )} - 1 = cot^{2} (θ)

Subtract

cot^{2} (θ)

from both sides

\frac{sec ^{2} ( θ ) - 1}{1 - cos ^{2} ( θ )} - 1 - cot^{2} (θ) = 0

Simplify

\frac{sec ^{2} ( θ ) - 1}{1 - cos ^{2} ( θ )} - 1 - cot^{2} (θ) : \frac{sec ^{2} ( θ ) + cos ^{2} ( θ ) + cot ^{2} ( θ ) cos ^{2} ( θ ) - cot ^{2} ( θ ) - 2}{1 - cos ^{2} ( θ )}

\frac{sec ^{2} ( θ ) - 1}{1 - cos ^{2} ( θ )} - 1 - cot^{2} (θ)

Convert element to fraction:

1 = \frac{1 ( 1 - cos ^{2} ( θ ) )}{1 - cos ^{2} ( θ )}, cot^{2} (θ) = \frac{cot ^{2} ( θ ) ( 1 - cos ^{2} ( θ ) )}{1 - cos ^{2} ( θ )}

= \frac{sec ^{2} ( θ ) - 1}{1 - cos ^{2} ( θ )} - \frac{1 \cdot ( 1 - cos ^{2} ( θ ) )}{1 - cos ^{2} ( θ )} - \frac{cot ^{2} ( θ ) ( 1 - cos ^{2} ( θ ) )}{1 - cos ^{2} ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{sec ^{2} ( θ ) - 1 - 1 \cdot ( 1 - cos ^{2} ( θ ) ) - cot ^{2} ( θ ) ( 1 - cos ^{2} ( θ ) )}{1 - cos ^{2} ( θ )}

Multiply:

1 \cdot (1 - cos^{2} (θ)) = (1 - cos^{2} (θ))

= \frac{sec ^{2} ( θ ) - 1 - ( - cos ^{2} ( θ ) + 1 ) - cot ^{2} ( θ ) ( - cos ^{2} ( θ ) + 1 )}{1 - cos ^{2} ( θ )}

Expand

sec^{2} (θ) - 1 - (1 - cos^{2} (θ)) - cot^{2} (θ) (1 - cos^{2} (θ)) : sec^{2} (θ) + cos^{2} (θ) + cot^{2} (θ) cos^{2} (θ) - cot^{2} (θ) - 2

sec^{2} (θ) - 1 - (1 - cos^{2} (θ)) - cot^{2} (θ) (1 - cos^{2} (θ))

- (1 - cos^{2} (θ)) : - 1 + cos^{2} (θ)

- (1 - cos^{2} (θ))

Distribute parentheses

= - (1) - (- cos^{2} (θ))

Apply minus-plus rules

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

= - 1 + cos^{2} (θ)

= sec^{2} (θ) - 1 - 1 + cos^{2} (θ) - cot^{2} (θ) (1 - cos^{2} (θ))

Expand

- cot^{2} (θ) (1 - cos^{2} (θ)) : - cot^{2} (θ) + cot^{2} (θ) cos^{2} (θ)

- cot^{2} (θ) (1 - cos^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - cot^{2} (θ), b = 1, c = cos^{2} (θ)

= - cot^{2} (θ) \cdot 1 - (- cot^{2} (θ)) cos^{2} (θ)

Apply minus-plus rules

- (- a) = a

= - 1 \cdot cot^{2} (θ) + cot^{2} (θ) cos^{2} (θ)

Multiply:

1 \cdot cot^{2} (θ) = cot^{2} (θ)

= - cot^{2} (θ) + cot^{2} (θ) cos^{2} (θ)

= sec^{2} (θ) - 1 - 1 + cos^{2} (θ) - cot^{2} (θ) + cot^{2} (θ) cos^{2} (θ)

Subtract the numbers:

- 1 - 1 = - 2

= sec^{2} (θ) + cos^{2} (θ) + cot^{2} (θ) cos^{2} (θ) - cot^{2} (θ) - 2

= \frac{sec ^{2} ( θ ) + cos ^{2} ( θ ) + cot ^{2} ( θ ) cos ^{2} ( θ ) - cot ^{2} ( θ ) - 2}{1 - cos ^{2} ( θ )}

\frac{sec ^{2} ( θ ) + cos ^{2} ( θ ) + cot ^{2} ( θ ) cos ^{2} ( θ ) - cot ^{2} ( θ ) - 2}{1 - cos ^{2} ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sec^{2} (θ) + cos^{2} (θ) + cot^{2} (θ) cos^{2} (θ) - cot^{2} (θ) - 2 = 0

Express with sin, cos

- 2 + cos^{2} (θ) - cot^{2} (θ) + sec^{2} (θ) + cos^{2} (θ) cot^{2} (θ)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - 2 + cos^{2} (θ) - (\frac{cos ( θ )}{sin ( θ )})^{2} + sec^{2} (θ) + cos^{2} (θ) (\frac{cos ( θ )}{sin ( θ )})^{2}

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= - 2 + cos^{2} (θ) - (\frac{cos ( θ )}{sin ( θ )})^{2} + (\frac{1}{cos ( θ )})^{2} + cos^{2} (θ) (\frac{cos ( θ )}{sin ( θ )})^{2}

Simplify

- 2 + cos^{2} (θ) - (\frac{cos ( θ )}{sin ( θ )})^{2} + (\frac{1}{cos ( θ )})^{2} + cos^{2} (θ) (\frac{cos ( θ )}{sin ( θ )})^{2} : \frac{- 2 sin ^{2} ( θ ) cos ^{2} ( θ ) + cos ^{4} ( θ ) sin ^{2} ( θ ) + cos ^{2} ( θ ) ( - cos ^{2} ( θ ) + cos ^{4} ( θ ) ) + sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

- 2 + cos^{2} (θ) - (\frac{cos ( θ )}{sin ( θ )})^{2} + (\frac{1}{cos ( θ )})^{2} + cos^{2} (θ) (\frac{cos ( θ )}{sin ( θ )})^{2}

(\frac{cos ( θ )}{sin ( θ )})^{2} = \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

(\frac{cos ( θ )}{sin ( θ )})^{2}

Apply exponent rule:

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

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2} = \frac{1}{cos ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

cos^{2} (θ) (\frac{cos ( θ )}{sin ( θ )})^{2} = \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )}

cos^{2} (θ) (\frac{cos ( θ )}{sin ( θ )})^{2}

(\frac{cos ( θ )}{sin ( θ )})^{2} = \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

(\frac{cos ( θ )}{sin ( θ )})^{2}

Apply exponent rule:

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

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

= \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} cos^{2} (θ)

Multiply fractions:

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

= \frac{cos ^{2} ( θ ) cos ^{2} ( θ )}{sin ^{2} ( θ )}

cos^{2} (θ) cos^{2} (θ) = cos^{4} (θ)

cos^{2} (θ) cos^{2} (θ)

Apply exponent rule:

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

cos^{2} (θ) cos^{2} (θ) = cos^{2 + 2} (θ)

= cos^{2 + 2} (θ)

Add the numbers:

2 + 2 = 4

= cos^{4} (θ)

= \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )}

= - 2 + cos^{2} (θ) - \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} + \frac{1}{cos ^{2} ( θ )} + \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )}

Combine the fractions

- \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} + \frac{cos ^{4} ( θ )}{sin ^{2} ( θ )} : \frac{- cos ^{2} ( θ ) + cos ^{4} ( θ )}{sin ^{2} ( θ )}

Apply rule

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

= \frac{- cos ^{2} ( θ ) + cos ^{4} ( θ )}{sin ^{2} ( θ )}

= - 2 + cos^{2} (θ) + \frac{cos ^{4} ( θ ) - cos ^{2} ( θ )}{sin ^{2} ( θ )} + \frac{1}{cos ^{2} ( θ )}

Convert element to fraction:

2 = \frac{2}{1}, cos^{2} (θ) = \frac{cos ^{2} ( θ )}{1}

= - \frac{2}{1} + \frac{cos ^{2} ( θ )}{1} + \frac{- cos ^{2} ( θ ) + cos ^{4} ( θ )}{sin ^{2} ( θ )} + \frac{1}{cos ^{2} ( θ )}

Least Common Multiplier of

1, 1, sin^{2} (θ), cos^{2} (θ) : sin^{2} (θ) cos^{2} (θ)

1, 1, sin^{2} (θ), cos^{2} (θ)

Lowest Common Multiplier (LCM)

Least Common Multiplier of

1, 1 : 1

1, 1

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

1

Multiply each factor the greatest number of times it occurs in either

1

1

= 1

Multiply the numbers:

1 = 1

= 1

Compute an expression comprised of factors that appear in at least one of the factored expressions

= sin^{2} (θ) cos^{2} (θ)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin^{2} (θ) cos^{2} (θ)

For

\frac{2}{1} :

multiply the denominator and numerator by

sin^{2} (θ) cos^{2} (θ)

\frac{2}{1} = \frac{2 sin ^{2} ( θ ) cos ^{2} ( θ )}{1 \cdot sin ^{2} ( θ ) cos ^{2} ( θ )} = \frac{2 sin ^{2} ( θ ) cos ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

For

\frac{cos ^{2} ( θ )}{1} :

multiply the denominator and numerator by

sin^{2} (θ) cos^{2} (θ)

\frac{cos ^{2} ( θ )}{1} = \frac{cos ^{2} ( θ ) sin ^{2} ( θ ) cos ^{2} ( θ )}{1 \cdot sin ^{2} ( θ ) cos ^{2} ( θ )} = \frac{cos ^{4} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

For

\frac{- cos ^{2} ( θ ) + cos ^{4} ( θ )}{sin ^{2} ( θ )} :

multiply the denominator and numerator by

cos^{2} (θ)

\frac{- cos ^{2} ( θ ) + cos ^{4} ( θ )}{sin ^{2} ( θ )} = \frac{( - cos ^{2} ( θ ) + cos ^{4} ( θ ) ) cos ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

For

\frac{1}{cos ^{2} ( θ )} :

multiply the denominator and numerator by

sin^{2} (θ)

\frac{1}{cos ^{2} ( θ )} = \frac{1 \cdot sin ^{2} ( θ )}{cos ^{2} ( θ ) sin ^{2} ( θ )} = \frac{sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

= - \frac{2 sin ^{2} ( θ ) cos ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )} + \frac{cos ^{4} ( θ ) sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )} + \frac{( - cos ^{2} ( θ ) + cos ^{4} ( θ ) ) cos ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )} + \frac{sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 sin ^{2} ( θ ) cos ^{2} ( θ ) + cos ^{4} ( θ ) sin ^{2} ( θ ) + ( - cos ^{2} ( θ ) + cos ^{4} ( θ ) ) cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

= \frac{- 2 sin ^{2} ( θ ) cos ^{2} ( θ ) + cos ^{4} ( θ ) sin ^{2} ( θ ) + cos ^{2} ( θ ) ( - cos ^{2} ( θ ) + cos ^{4} ( θ ) ) + sin ^{2} ( θ )}{sin ^{2} ( θ ) cos ^{2} ( θ )}

\frac{sin ^{2} ( θ ) + ( - cos ^{2} ( θ ) + cos ^{4} ( θ ) ) cos ^{2} ( θ ) + cos ^{4} ( θ ) sin ^{2} ( θ ) - 2 cos ^{2} ( θ ) sin ^{2} ( θ )}{cos ^{2} ( θ ) sin ^{2} ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin^{2} (θ) + (- cos^{2} (θ) + cos^{4} (θ)) cos^{2} (θ) + cos^{4} (θ) sin^{2} (θ) - 2 cos^{2} (θ) sin^{2} (θ) = 0

Factor

sin^{2} (θ) + (- cos^{2} (θ) + cos^{4} (θ)) cos^{2} (θ) + cos^{4} (θ) sin^{2} (θ) - 2 cos^{2} (θ) sin^{2} (θ) : (- 1 + cos (θ)) (1 + cos (θ)) (sin^{2} (θ) (1 + cos (θ)) (- 1 + cos (θ)) + cos^{4} (θ))

sin^{2} (θ) + (- cos^{2} (θ) + cos^{4} (θ)) cos^{2} (θ) + cos^{4} (θ) sin^{2} (θ) - 2 cos^{2} (θ) sin^{2} (θ)

(- cos^{2} (θ) + cos^{4} (θ)) cos^{2} (θ) = cos^{4} (θ) (cos (θ) + 1) (cos (θ) - 1)

(- cos^{2} (θ) + cos^{4} (θ)) cos^{2} (θ)

Factor

- cos^{2} (θ) + cos^{4} (θ) : cos^{2} (θ) (cos (θ) + 1) (cos (θ) - 1)

- cos^{2} (θ) + cos^{4} (θ)

Factor out common term

cos^{2} (θ) : cos^{2} (θ) (cos^{2} (θ) - 1)

cos^{4} (θ) - cos^{2} (θ)

Apply exponent rule:

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

cos^{4} (θ) = cos^{2} (θ) cos^{2} (θ)

= cos^{2} (θ) cos^{2} (θ) - cos^{2} (θ)

Factor out common term

cos^{2} (θ)

= cos^{2} (θ) (cos^{2} (θ) - 1)

= cos^{2} (θ) (cos^{2} (θ) - 1)

Factor

cos^{2} (θ) - 1 : (cos (θ) + 1) (cos (θ) - 1)

cos^{2} (θ) - 1

Rewrite

1

1^{2}

= cos^{2} (θ) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (θ) - 1^{2} = (cos (θ) + 1) (cos (θ) - 1)

= (cos (θ) + 1) (cos (θ) - 1)

= cos^{2} (θ) (cos (θ) + 1) (cos (θ) - 1)

= cos^{2} (θ) (cos (θ) + 1) (cos (θ) - 1) cos^{2} (θ)

Refine

= cos^{4} (θ) (cos (θ) + 1) (cos (θ) - 1)

= sin^{2} (θ) + cos^{4} (θ) (cos (θ) + 1) (cos (θ) - 1) + cos^{4} (θ) sin^{2} (θ) - 2 cos^{2} (θ) sin^{2} (θ)

Factor out common term

sin^{2} (θ)

= sin^{2} (θ) (1 + cos^{4} (θ) - 2 cos^{2} (θ)) + cos^{4} (θ) (1 + cos (θ)) (- 1 + cos (θ))

Factor

cos^{4} (θ) - 2 cos^{2} (θ) + 1 : (cos (θ) + 1)^{2} (cos (θ) - 1)^{2}

1 + cos^{4} (θ) - 2 cos^{2} (θ)

Let

u = cos^{2} (θ)

= u^{2} - 2 u + 1

Factor

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

u^{2} - 2 u + 1

Break the expression into groups

u^{2} - 2 u + 1

Definition

Factors of

1 : 1

1

Divisors (Factors)

Find the Prime factors of

1 :

Add 1

1

The factors of

1

1

Negative factors of

1 : - 1

Multiply the factors by

- 1

to get the negative factors

- 1

For every two factors such that

u * v = 1,

check if

u + v = - 2

Check

u = 1, v = 1 : u * v = 1, u + v = 2 \Rightarrow

FalseCheck

u = - 1, v = - 1 : u * v = 1, u + v = - 2 \Rightarrow

True

u = - 1, v = - 1

Group into

(a x^{2} + ux) + (vx + c)

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

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

Factor out

u

from

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

u^{2} - u

Apply exponent rule:

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

u^{2} = uu

= uu - u

Factor out common term

u

= u (u - 1)

Factor out

- 1

from

- u + 1 : - (u - 1)

- u + 1

Factor out common term

- 1

= - (u - 1)

= u (u - 1) - (u - 1)

Factor out common term

u - 1

= (u - 1) (u - 1)

= (u - 1) (u - 1)

Substitute back

u = cos^{2} (θ)

= (cos^{2} (θ) - 1) (cos^{2} (θ) - 1)

Factor

cos^{2} (θ) - 1 : (cos (θ) + 1) (cos (θ) - 1)

cos^{2} (θ) - 1

Rewrite

1

1^{2}

= cos^{2} (θ) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (θ) - 1^{2} = (cos (θ) + 1) (cos (θ) - 1)

= (cos (θ) + 1) (cos (θ) - 1)

= (cos (θ) + 1) (cos (θ) - 1) (cos^{2} (θ) - 1)

Factor

cos^{2} (θ) - 1 : (cos (θ) + 1) (cos (θ) - 1)

cos^{2} (θ) - 1

Rewrite

1

1^{2}

= cos^{2} (θ) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (θ) - 1^{2} = (cos (θ) + 1) (cos (θ) - 1)

= (cos (θ) + 1) (cos (θ) - 1)

= (cos (θ) + 1) (cos (θ) - 1) (cos (θ) + 1) (cos (θ) - 1)

Refine

= (cos (θ) + 1)^{2} (cos (θ) - 1)^{2}

= cos^{4} (θ) (cos (θ) + 1) (cos (θ) - 1) + sin^{2} (θ) (cos (θ) + 1)^{2} (cos (θ) - 1)^{2}

Apply exponent rule:

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

(1 + cos (θ))^{2} = (1 + cos (θ)) (1 + cos (θ)), (- 1 + cos (θ))^{2} = (- 1 + cos (θ)) (- 1 + cos (θ))

= sin^{2} (θ) (1 + cos (θ)) (1 + cos (θ)) (- 1 + cos (θ)) (- 1 + cos (θ)) + cos^{4} (θ) (1 + cos (θ)) (- 1 + cos (θ))

Factor out common term

(- 1 + cos (θ)) (1 + cos (θ))

= (- 1 + cos (θ)) (1 + cos (θ)) (sin^{2} (θ) (1 + cos (θ)) (- 1 + cos (θ)) + cos^{4} (θ))

(- 1 + cos (θ)) (1 + cos (θ)) (sin^{2} (θ) (1 + cos (θ)) (- 1 + cos (θ)) + cos^{4} (θ)) = 0

Solving each part separately

- 1 + cos (θ) = 0 or 1 + cos (θ) = 0 or sin^{2} (θ) (1 + cos (θ)) (- 1 + cos (θ)) + cos^{4} (θ) = 0

- 1 + cos (θ) = 0 : θ = 2 πn

- 1 + cos (θ) = 0

Move

1

to the right side

- 1 + cos (θ) = 0

Add

1

to both sides

- 1 + cos (θ) + 1 = 0 + 1

Simplify

cos (θ) = 1

cos (θ) = 1

General solutions for

cos (θ) = 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

θ = 0 + 2 πn

θ = 0 + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn

1 + cos (θ) = 0 : θ = π + 2 πn

1 + cos (θ) = 0

Move

1

to the right side

1 + cos (θ) = 0

Subtract

1

from both sides

1 + cos (θ) - 1 = 0 - 1

Simplify

cos (θ) = - 1

cos (θ) = - 1

General solutions for

cos (θ) = - 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

θ = π + 2 πn

θ = π + 2 πn

sin^{2} (θ) (1 + cos (θ)) (- 1 + cos (θ)) + cos^{4} (θ) = 0 : θ = arccos (\frac{1}{2}) + 2 πn, θ = 2 π - arccos (\frac{1}{2}) + 2 πn, θ = arccos (- \frac{1}{2}) + 2 πn, θ = - arccos (- \frac{1}{2}) + 2 πn

sin^{2} (θ) (1 + cos (θ)) (- 1 + cos (θ)) + cos^{4} (θ) = 0

Rewrite using trig identities

cos^{4} (θ) + (- 1 + cos (θ)) (1 + cos (θ)) sin^{2} (θ)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= cos^{4} (θ) + (- 1 + cos (θ)) (1 + cos (θ)) (1 - cos^{2} (θ))

Simplify

cos^{4} (θ) + (- 1 + cos (θ)) (1 + cos (θ)) (1 - cos^{2} (θ)) : 2 cos^{2} (θ) - 1

cos^{4} (θ) + (- 1 + cos (θ)) (1 + cos (θ)) (1 - cos^{2} (θ))

Expand

(- 1 + cos (θ)) (1 + cos (θ)) (1 - cos^{2} (θ)) : 2 cos^{2} (θ) - cos^{4} (θ) - 1

Expand

(- 1 + cos (θ)) (1 + cos (θ)) : cos^{2} (θ) - 1

(- 1 + cos (θ)) (1 + cos (θ))

Apply Difference of Two Squares Formula:

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

a = cos (θ), b = 1

= cos^{2} (θ) - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= cos^{2} (θ) - 1

= (cos^{2} (θ) - 1) (1 - cos^{2} (θ))

Expand

(cos^{2} (θ) - 1) (1 - cos^{2} (θ)) : 2 cos^{2} (θ) - cos^{4} (θ) - 1

(cos^{2} (θ) - 1) (1 - cos^{2} (θ))

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = cos^{2} (θ), b = - 1, c = 1, d = - cos^{2} (θ)

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

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

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

Simplify

1 \cdot cos^{2} (θ) - cos^{2} (θ) cos^{2} (θ) - 1 \cdot 1 + 1 \cdot cos^{2} (θ) : 2 cos^{2} (θ) - cos^{4} (θ) - 1

1 \cdot cos^{2} (θ) - cos^{2} (θ) cos^{2} (θ) - 1 \cdot 1 + 1 \cdot cos^{2} (θ)

Group like terms

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

Add similar elements:

1 \cdot cos^{2} (θ) + 1 \cdot cos^{2} (θ) = 2 cos^{2} (θ)

= 2 cos^{2} (θ) - cos^{2} (θ) cos^{2} (θ) - 1 \cdot 1

cos^{2} (θ) cos^{2} (θ) = cos^{4} (θ)

cos^{2} (θ) cos^{2} (θ)

Apply exponent rule:

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

cos^{2} (θ) cos^{2} (θ) = cos^{2 + 2} (θ)

= cos^{2 + 2} (θ)

Add the numbers:

2 + 2 = 4

= cos^{4} (θ)

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= 2 cos^{2} (θ) - cos^{4} (θ) - 1

= 2 cos^{2} (θ) - cos^{4} (θ) - 1

= 2 cos^{2} (θ) - cos^{4} (θ) - 1

= cos^{4} (θ) + 2 cos^{2} (θ) - cos^{4} (θ) - 1

Add similar elements:

cos^{4} (θ) - cos^{4} (θ) = 0

= 2 cos^{2} (θ) - 1

= 2 cos^{2} (θ) - 1

- 1 + 2 cos^{2} (θ) = 0

Solve by substitution

- 1 + 2 cos^{2} (θ) = 0

Let:

cos (θ) = u

- 1 + 2 u^{2} = 0

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

- 1 + 2 u^{2} = 0

Move

1

to the right side

- 1 + 2 u^{2} = 0

Add

1

to both sides

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

Simplify

2 u^{2} = 1

2 u^{2} = 1

Divide both sides by

2

2 u^{2} = 1

Divide both sides by

2

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = cos (θ)

cos (θ) = \frac{1}{2}, cos (θ) = - \frac{1}{2}

cos (θ) = \frac{1}{2}, cos (θ) = - \frac{1}{2}

cos (θ) = \frac{1}{2} : θ = arccos (\frac{1}{2}) + 2 πn, θ = 2 π - arccos (\frac{1}{2}) + 2 πn

cos (θ) = \frac{1}{2}

Apply trig inverse properties

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \frac{1}{2}

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

θ = arccos (\frac{1}{2}) + 2 πn, θ = 2 π - arccos (\frac{1}{2}) + 2 πn

θ = arccos (\frac{1}{2}) + 2 πn, θ = 2 π - arccos (\frac{1}{2}) + 2 πn

cos (θ) = - \frac{1}{2} : θ = arccos (- \frac{1}{2}) + 2 πn, θ = - arccos (- \frac{1}{2}) + 2 πn

cos (θ) = - \frac{1}{2}

Apply trig inverse properties

cos (θ) = - \frac{1}{2}

General solutions for

cos (θ) = - \frac{1}{2}

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

θ = arccos (- \frac{1}{2}) + 2 πn, θ = - arccos (- \frac{1}{2}) + 2 πn

θ = arccos (- \frac{1}{2}) + 2 πn, θ = - arccos (- \frac{1}{2}) + 2 πn

Combine all the solutions

θ = arccos (\frac{1}{2}) + 2 πn, θ = 2 π - arccos (\frac{1}{2}) + 2 πn, θ = arccos (- \frac{1}{2}) + 2 πn, θ = - arccos (- \frac{1}{2}) + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = arccos (\frac{1}{2}) + 2 πn, θ = 2 π - arccos (\frac{1}{2}) + 2 πn, θ = arccos (- \frac{1}{2}) + 2 πn, θ = - arccos (- \frac{1}{2}) + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

θ = arccos (\frac{1}{2}) + 2 πn, θ = 2 π - arccos (\frac{1}{2}) + 2 πn, θ = arccos (- \frac{1}{2}) + 2 πn, θ = - arccos (- \frac{1}{2}) + 2 πn

Show solutions in decimal form

θ = 0.78539 \dots + 2 πn, θ = 2 π - 0.78539 \dots + 2 πn, θ = 2.35619 \dots + 2 πn, θ = - 2.35619 \dots + 2 πn

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