sin^2(θ)-cos^2(θ)=1+cos(θ)

Solution

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

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

Solution steps

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

Subtract

1 + cos (θ)

from both sides

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

Rewrite using trig identities

- cos (θ) - 2 cos^{2} (θ) = 0

Solve by substitution

cos (θ) = - \frac{1}{2}, cos (θ) = 0

cos (θ) = - \frac{1}{2} : θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

cos (θ) = 0 : θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

Combine all the solutions

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

cos (x) sin (x) - 2 cos (x) = 0

csc (x) = - 2

cos^{2} (x) - sin^{2} (x) = 0

tan (θ) = 2 sin (θ)

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