Evaluate the definite integral $$\int_0^\pi\bigg(\frac{\sin(2x)\sin(3x)\sin(5x)\sin(30x)}{\sin x\sin(6x)\sin(10x)\sin(15x)}\bigg)^2dx$$
My Trial:
Using double angle formula, $$\frac{\sin(2x)\sin(3x)\sin(5x)\sin(30x)}{\sin x\sin(6x)\sin(10x)\sin(15x)}=\frac{\cos x\cos(15x)}{\cos(3x)\cos(5x)}$$ We have $\cos x\cos(15x)=\frac12[\cos(16x)+\cos(14x)]$ and $\cos (3x)\cos(5x)=\frac12[\cos(8x)+\cos(2x)]$
But that doesn't seem to give me any help.
