Question

Prove that, $\frac{\cos (7x-5y)+\cos (7y-5x)}{\sin (7x-5y)+\sin (7y-5x)}=\cot (x+y)$.

Answer 1

$$\begin{array}{c}=\frac{2\cos \left(\frac{2x+2y}{2}\right)\cdot \cos (12x-12y)}{2\sin \left(\frac{2x+2y}{2}\right)\cdot \cos (12x-12y)}\\ =\frac{\cos 2\frac{\left(x+y\right)}{2}}{\sin 2\frac{\left(x+y\right)}{2}}\\ =\frac{\cos \left(x+y\right)}{\sin \left(x+y\right)}=\cot \left(x+y\right)\end{array}$$