What is the general solution for 2cos(θ/2)=cos(θ)+1 ?

The general solution for 2cos(θ/2)=cos(θ)+1 is θ=pi+4pin,θ=3pi+4pin,θ=4pin

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2cos(θ/2)=cos(θ)+1

2 cos (\frac{θ}{2}) = cos (θ) + 1

θ = π + 4 πn, θ = 3 π + 4 πn, θ = 4 πn

Solution steps

2 cos (\frac{θ}{2}) = cos (θ) + 1

Subtract

cos (θ) + 1

from both sides

2 cos (\frac{θ}{2}) - cos (θ) - 1 = 0

Let:

u = \frac{θ}{2}

2 cos (u) - cos (2 u) - 1 = 0

Rewrite using trig identities

2 cos (u) - 2 cos^{2} (u) = 0

Solve by substitution

cos (u) = 0, cos (u) = 1

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

cos (u) = 1 : u = 2 πn

Combine all the solutions

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = 2 πn

Substitute back

u = \frac{θ}{2}

\frac{θ}{2} = \frac{π}{2} + 2 πn : θ = π + 4 πn

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

\frac{θ}{2} = 2 πn : θ = 4 πn

θ = π + 4 πn, θ = 3 π + 4 πn, θ = 4 πn

cos (3 x) + sin (3 x) = 0

\frac{1}{3} = tan (x)

sin (θ) = - \frac{5}{5}

0.9 = cos (x)

cos (x) = \frac{8}{9}

What is the general solution for 2cos(θ/2)=cos(θ)+1 ?
The general solution for 2cos(θ/2)=cos(θ)+1 is θ=pi+4pin,θ=3pi+4pin,θ=4pin