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Math Help - simplify using identities

  1. #1
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    simplify using identities

    \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,
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  2. #2
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    (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)
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  3. #3
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    Thanks! I think I understand it now. So, for future reference, I'm allowed to cross multiply when proving an identity?
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  4. #4
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    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 ys 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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