Find the value of m parameter such that the equation has solutions

Question

I have the following equation $cos^{3}(x)\cdot sin(x) - sin^{3}(x)\cdot cos(x)=m$

I need to find $m$ such that the equation has solutions.The right answer is $[-1/4,1/4]$

I tried with Rolle's Theorem.The derivative of f(x) is cos4x but I get stuck with cos4x=0 because I don't get some exact values.There's other way to solve it?

Answer 1

Hint: $$ \cos^3x\sin x - \sin^3 x\cos x= \sin x \cos x(\cos^2x-\sin^2x) = \frac{1}{2}\sin 2x \cos 2x = \frac{1}{4}\sin 4x. $$