Help on Precalculus problem

Question

cos4xcosx - sin4xsinx = 0 in the interval [0,2pi]

Thank you!!!

Answer 1

Recall the cosine angle addition rule

cos(A + B) = cos A cos B - sin A sin B

Set A = 4x and B = x to obtain

cos 5x = cos 4x cos x - sin 4x sin x

Your equation is now reduced to cos 5x = 0. The zeros of the cosine in the interval [0, 10π] are

π/2, 3π/2, 5π/2, 7π/2, 9π/2, 11π/2, 13π/2, 15π/2, 17π/2, 19π/2

Dividing by 5 gives the x-solutions in [0,2π]:

x = π/10, 3π/10, π/2, 7π/10, 9π/10, 11π/10, 13π/10, 3π/2, 17π/10, 19π/10