<s>[INST] 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.
 [/INST] $$\sum_{n=1}^{N} {\sin(nx)} = \frac{1}{2} \cot{\frac{x}{2}} - \frac{\cos{(N+\frac{1}{2})x}}{2\sin{\frac{x}{2}}}$$
$$\sum_{n=1}^{44} {\sin(nx)} = \frac{1}{2} \cot{\frac{x}{2}} - \frac{\cos{(44+\frac{1}{2})x}}{2\sin{\frac{x}{2}}}$$
$$\sum_{n=1}^{44} {\sin(2 n)} = \frac{1}{2} \cot{\frac{2}{2}} - \frac{\cos{[(44+\frac{1}{2})2]}}{2\sin{\frac{2}{2}}}$$
$$\sum_{n=1}^{44} {\sin(2 n)} = \frac{1}{2} \cot{1} - \frac{1}{2}\frac{\cos{99}}{\sin{1}}$$
$$\sum_{n=1}^{44} {\sin(2 n)} \approx 0.2974$$
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Lagrange.27s_trigonometric_identities
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