Solve the exponential-trigonometry equation

Question

I have the problem $$ 3^{\sin x} \cdot 3^{\sin 2x} \cdot 3^{\sin 3x} \cdot \ldots = 3. $$

I’ve converted it to $$ \sin x + \sin 2x + \sin 3x + \ldots = 1, $$ but what should I do next?

Answer 1

Hint: Use that for your sum is hold: $$\sum_{i=1}^{n}\sin(ix)=\csc \left(\frac{x}{2}\right) \sin \left(\frac{n x}{2}\right) \sin \left(\frac{1}{2} (n+1) x\right)$$