Question:
If $x = \sin 1^\circ$, find the value of the expression: $$\frac{1}{\cos0^\circ \cos1^\circ} + \frac{1}{\cos1^\circ\cos2^\circ} + ... + \frac{1}{\cos44^\circ\cos45^\circ}$$ in terms of $x$
I really can't see how I would simplify this expression in terms of $x$. Any hint would be appreciated.
Hint $$\sin{1}=\sin{[(n+1)-n)]}=\sin{(n+1)}\cos{n}-\cos{(n+1)}\sin{n}$$ so \begin{align}\sum_{n=0}^{44}\dfrac{1}{\cos{n}\cos{(n+1)}} &=\dfrac{1}{\sin{1}}\sum_{n=0}^{45}\dfrac{\sin{(n+1)}\cos{n}-\cos{(n+1)}\sin{n}}{\cos{n}\cos{(n+1)}}\\ \ \\ &=\frac1 {\sin1}\,\sum_{n=0}^{44}[\tan{(n+1)}-\tan{n}] \end{align}
