Question

$\sum_{h=1}^{n} \frac{cos(3n+1)x - cos(3n+2)}{1+2cos(2n+1)x}$

Answer 1

$\displaystyle \frac{1}{2}\left[\tan\frac{(2n+1)x}{2}-\tan\frac{x}{2}\right]$

Answer 2

Rewrite the numerator using the cosine subtraction formula and express it, along with the denominator, in a form that can be written as the difference of two tangents divided by 2. The sum telescopes, leaving only the first and last terms.