When are y equals sin x and y equals cos x times cos x equal?

y=sin x

y=cos x cos x

sin x = cos^2 x

sin x = 1-sin^2 x

sin x -1 + sin^2 x = 0

sin^2 x + sin x -1 = 0

Let y=sin x

y^2+y-1 = 0

This equation is of form ay^2+by+c=0

a = 1 b = 1 c = -1

y=[-b+/-sqrt(b^2-4ac)]/2a]

y=[-1 +/-sqrt(1^2-4(1)(-1)]/(2)(1)

discriminant is b^2-4ac =5

y=[-1 +√(5)] / 2

y=[-1 -√(5)] / 2

sin x = [-1 +√(5)] / 2

x = sin^-1 [-1 +√(5)] / 2] = 0.6662394 radians

x = sin^-1 [-1 -√(5)] / 2] = sin^-1 (-1.618) -- has no solution

When x = 0.6662394 radians, sin x and cos x times cos x are equal.