What is the general solution for 6cos(2θ)=6cos(θ) ?

The general solution for 6cos(2θ)=6cos(θ) is θ=2pin,θ=(2pi)/3+2pin,θ=(4pi)/3+2pin

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6cos(2θ)=6cos(θ)

6 cos (2 θ) = 6 cos (θ)

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

Solution steps

6 cos (2 θ) = 6 cos (θ)

Subtract

6 cos (θ)

from both sides

6 cos (2 θ) - 6 cos (θ) = 0

Rewrite using trig identities

(- 1 + 2 cos^{2} (θ)) \cdot 6 - 6 cos (θ) = 0

Solve by substitution

cos (θ) = 1, cos (θ) = - \frac{1}{2}

cos (θ) = 1 : θ = 2 πn

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

Combine all the solutions

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

tan (\frac{θ}{2}) = csc^{2} (θ) - cot^{2} (θ)

6 cos^{2} (x) - sin (x) - 4 = 0

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

2 cos (θ) - 1 = 0, 0 \leq θ \leq 2 π

3 - 3 cos (x) = 2 sin^{2} (x)

What is the general solution for 6cos(2θ)=6cos(θ) ?
The general solution for 6cos(2θ)=6cos(θ) is θ=2pin,θ=(2pi)/3+2pin,θ=(4pi)/3+2pin