Find $4\sin (x)\,\big(\sin(3x)+\sin (7x)+\sin(11x)+\sin(15x)\big)$ when $x=\frac{\pi}{34}$.

Question

If $x=\frac{\pi}{34}$, then find the value of
$$S=4\sin (x)\,\big(\sin(3x)+\sin (7x)+\sin(11x)+\sin(15x)\big).$$

Source: Joszef Wildt International Math Competition Problem

My attempt:

$$\sin(3x)+\sin(15x)=2\sin(9x)\cos(6x)$$

$$\sin(7x)+\sin(11x)=2\sin(9x)\cos(2x)$$

So $$S=8\sin(x)\sin(9x)\big(\cos(6x)+\cos(2x)\big)$$

$$S=16\sin(x)\sin(9x)\cos(4x)\cos(2x)$$
$$S=16\sin(x)\sin(9x)\sin(13x)\sin(15x)$$

any clue here?

Answer 1

Alternatively, telescope the expression with

\begin{align} &4\sin x(\sin 3x+\sin 7x+\sin 11x+\sin 15x)\\ =&\frac{2\sin2x}{\cos x}(\sin 3x+\sin 7x+\sin 11x+\sin 15x)\\ =&\frac 1{\cos x} [ (\cos x-\cos5x) + (\cos 5x-\cos9x) + (\cos 9x-\cos13x) + (\cos 13x-\cos17x)]\\ =& \frac{1}{\cos x}(\cos x - \cos 17x) =1-\frac{\cos\frac\pi2}{\cos\frac{\pi}{34}}=1 \end{align}