simplify using identities

**absvalue** · October 27th 2009, 12:52 AM

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

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.

Thanks for your help,

**james_bond** · October 27th 2009, 01:17 AM

$(1-2\cos^2y)(\sin y-\cos y)=$ $(\sin y+ \cos y)(1-2\cos y\sin y)\Leftrightarrow \sin y-\cos y-2\cos^2y\sin y+2\cos^3y=$ $\sin y+ \cos y-2\cos^2 y\sin y-2\cos y\sin^2 y$ $\Leftrightarrow \cos^3y=\cos y-\cos y\sin^2y=\cos y (1-\sin^2y)$

**absvalue** · October 27th 2009, 01:39 AM

Thanks! I think I understand it now. So, for future reference, I'm allowed to cross multiply when proving an identity?

**james_bond** · October 27th 2009, 03:55 AM

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

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