Q: Simplify the trigonometric expression Simplify the expression $$\left(1-\frac{\cos61^{\circ}}{\cos1^{\circ}}\right) \left(1-\frac{\cos62^{\circ}}{\cos2^{\circ}}\right)\cdot ...\cdot \left(1-\frac{\cos119^{\circ}}{\cos59^{\circ}}\right)$$

A: We have
$$\begin{align}\prod_{i=1}^{59}\left(1-\frac{\cos(60^\circ+i^\circ)}{\cos i^\circ}\right)&=\prod_{i=1}^{59}\frac{\cos i^\circ-\cos (60^\circ+i^\circ)}{\cos i^\circ}\\&=\prod_{i=1}^{59}\frac{-2\sin(30^\circ+i^\circ)\sin (-30^\circ)}{\cos i^\circ}\\&=\prod_{i=1}^{59}\frac{\sin(30^\circ+i^\circ)}{\cos i^\circ}\\&\color{red}{=}\prod_{i=1}^{59}\frac{\sin(90^\circ-i^\circ)}{\cos i^\circ}\\&=\prod_{i=1}^{59}1=1.\end{align}$$
The equality "$\color{red}{=}$" comes from that the numerator is
$$\sin31^\circ\sin 32^\circ\cdots\sin89^\circ=\sin89^\circ\sin88^\circ\cdots\sin31^\circ.$$

A: Hint 
Reduce each piece of the product to the same denominator and use $$\cos(p)-\cos(q)=-2\sin[(p+q)/2] \sin[(p-q)/2]$$ Some nice things would come out.
I am sure that you can take from here.
