Solve X for Sinx= Cos5x

To solve this equation, first we have to recall the fact that sine and cosine are complementary functions, so that:

cos x = sin (pi/2 - x) and sin x = cos (pi/2 - x) 

We'll substitute sin x by  cos (pi/2 - x), so that:

cos (pi/2 - x) = cos5x

cos (pi/2 - x) - cos5x = 0

We'll transform the difference into a product:

2 sin [(pi/2 - x + x)/2]sin[(pi/2 - x - x)/2] = 0

2sin (pi/4)sin(pi/4 - x) = 0

We'll divide by 2sin (pi/4) and we'll get:

sin(pi/4 - x) = 0

This is an elementary equation:

pi/4 - x = (-1)^k*arcsin0 + k*pi

-x = -pi/4 + k*pi

We'll multiply by -1:

x = pi/4 - k*pi