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Popular >

(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

+1

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 ) + 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

\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))

(- 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 : 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

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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