How would you solve cos (5x) = sin (10x) ?

We will use double angle identities:

cos (5x) = sin (10x)

cos (5x) = 2 cos (5x) sin (5x)

cos (5 x) - 2 cos (5 x) sin (5x) = 0

cos (5 x) · [ 1 - 2 sin (5 x) ] = 0

cos (5 x) = 0 or : 1 - 2 sin (5 x) = 0

5 x = π/2 + kπ, k∈Z sin (5 x) = 1/2

x1 = π/10 + kπ/5 5 x = π/6+2kπ, k∈ Z

5 x = 5π/6 + 2kπ, k∈ Z

x 2 = π/30 + 2kπ/5

x 3 = π/9 + 2kπ/5