trigonometry - Simplify a quick sum of sines

Simplify $\sin 2+\sin 4+\sin 6+\cdots+\sin 88$

I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.

Answer

The angles are in arithmetic progression. Use the formula

$$\sum_{k=0}^{n-1} \sin (a+kb) = \frac{\sin \frac{nb}{2}}{\sin \frac{b}{2}} \sin \left( a+ (n-1)\frac{b}{2}\right)$$

See here for two proofs (using trigonometry, or using complex numbers).

In your case, $a=b=2$ and $n=44$.