<s>[INST] Find $4\sin (x)\,\big(\sin(3x)+\sin (7x)+\sin(11x)+\sin(15x)\big)$ when $x=\frac{\pi}{34}$. 
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?
 [/INST] as You simplified and as @WE Tutorial School pointed out,
$$S=16\cos\frac{\pi}{17}\cos\frac{2\pi}{17}\cos\frac{4\pi}{17}\cos\frac{8\pi}{17}.$$ 
Applying $\sin 2x = 2 \sin x \cos x$ 4 times,
$$ S \cdot \sin \frac \pi{17}  = \sin \frac {16\pi}{17}$$
$$ S = \frac {\sin \left( \pi - \frac \pi{17} \right)}{\sin \frac \pi{17}}$$
$$S=1$$
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