What is the general solution for cos(x)=cos(x/2) ?

The general solution for cos(x)=cos(x/2) is x=(8pin)/3 ,x=(4pi)/3+(8pin)/3

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cos(x)=cos(x/2)

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

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

Solution steps

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

Subtract

cos (\frac{x}{2})

from both sides

cos (x) - cos (\frac{x}{2}) = 0

Rewrite using trig identities

- 2 sin (\frac{3 x}{4}) sin (\frac{x}{4}) = 0

Solving each part separately

sin (\frac{3 x}{4}) = 0 or sin (\frac{x}{4}) = 0

sin (\frac{3 x}{4}) = 0 : x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (\frac{x}{4}) = 0 : x = 8 πn, x = 4 π + 8 πn

Combine all the solutions

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

Merge Overlapping Intervals

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

3 tan^{2} (x) - 3 tan (x) = 0

- 4 sec^{2} (x) + 19 = - 4 tan (x)

4 sin (x) + 6 = 6

tan (2 x - \frac{π}{6}) = \frac{- 1}{3}

2 cos (x) + 2 sin (x) = 1

What is the general solution for cos(x)=cos(x/2) ?
The general solution for cos(x)=cos(x/2) is x=(8pin)/3 ,x=(4pi)/3+(8pin)/3