What is the general solution for cos(x)=cos(3x) ?

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

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cos(x)=cos(3x)

cos (x) = cos (3 x)

x = πn, x = \frac{π}{2} + πn

Solution steps

cos (x) = cos (3 x)

Subtract

cos (3 x)

from both sides

cos (x) - cos (3 x) = 0

Rewrite using trig identities

2 sin (2 x) sin (x) = 0

Solving each part separately

sin (2 x) = 0 or sin (x) = 0

sin (2 x) = 0 : x = πn, x = \frac{π}{2} + πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

x = πn, x = \frac{π}{2} + πn, x = 2 πn, x = π + 2 πn

Merge Overlapping Intervals

x = πn, x = \frac{π}{2} + πn

sin (θ) = tan (θ)

sin (x) = \frac{1}{8}

2 cos (2 θ) = 1

8 cos (θ) - 43 = 0

cos (x) = 1 + sin (x)

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