1. ## simplify using identities

$\displaystyle \frac{1-2cos^2{y}}{1-2cos{y} \times sin{y}} = \frac{sin{y} + cos{y}}{sin{y} - cos{y}}$

Could someone give me a pointer? I've been banging my head against this problem for over an hour. I can't seem to find a way to simplify the left hand side.

2. $\displaystyle (1-2\cos^2y)(\sin y-\cos y)=$$\displaystyle (\sin y+ \cos y)(1-2\cos y\sin y)\Leftrightarrow \sin y-\cos y-2\cos^2y\sin y+2\cos^3y=$$\displaystyle \sin y+ \cos y-2\cos^2 y\sin y-2\cos y\sin^2 y$$\displaystyle \Leftrightarrow \cos^3y=\cos y-\cos y\sin^2y=\cos y (1-\sin^2y)$
4. Yeah in most of the times although you have to be careful. For example here the identity doesn't hold for $\displaystyle y=\pi /4$ because on both sides the denominator would be 0 but when we cross multiplied, the expression we got holds for $\displaystyle y=\pi /4$. The other manipulations were equivalent on the given domain therefore we proved the identity for all $\displaystyle y$s except $\displaystyle y=\pi /4+k\pi$ ($\displaystyle k\in\mathbb{Z}$) which we had to do. So in general you should check the domains.