Trignometry question find the given

Question

If $\cos A=3/4$ then what is the value of $32\sin(A/2)\sin(5A/2)$?

Answer 1

You surely know that $\sin\alpha\sin\beta=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta))$. With $\alpha=5A/2$ and $\beta=A/2$, you get $$ 32\sin\frac{5A}{2}\sin\frac{A}{2}= 16(\cos2A-\cos3A) $$ Now apply other known formulas: both $\cos2A$ and $\cos3A$ can be expressed in terms of $\cos A$.