Question

prove that sin theta + sin 5 theta + sin 9 theta + sin 13 theta /cos theta + cos 5 theta + cos 9 theta + cos 13 theta = tan7 theta​

Answer 1

$\frac{sin \theta + sin5 \theta + sin9 \theta + sin13 \theta}{cos \theta + cos5 \theta + cos9 \theta + cos13 \theta} \\ \\ = \frac{(sin13 \theta + sin \theta) + (sin 9\theta + sin5 \theta)}{(cos13 \theta + cos \theta) + (cos9 \theta + cos 5\theta)} \\ \\ = \frac{2sin \frac{14 \theta}{2}cos \frac{12 \theta}{2} + 2sin \frac{14 \theta}{2} cos \frac{4 \theta}{2} }{2cos \frac{14 \theta}{2}cos \frac{12 \theta}{2} + 2cos \frac{14 \theta}{2}cos \frac{4 \theta}{2} } \\ \\ = \frac{sin7 \theta cos6 \theta + sin7 \theta cos2 \theta}{cos7 \theta cos6 \theta + cos7 \theta cos2 \theta} \\ \\ = \frac{sin7 \theta}{cos7 \theta} \\ \\ = tan7 \theta$