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I need to verify this trigonometric identity for an assignment:$$\frac{(\sin x - \cos x + 1)}{(\sin x + \cos x-1)}=\frac{\sin x + 1}{\cos x}$$

I've tried a few different approaches, but I end up getting lost in messy equations and no immediately visible equivalence between the two sides. I'd appreciate some help!

Also, it's not necessery to give me the direct answer. Some helpful hints about which first steps to take would be better, so I can find the right direction to go in and then go on to solve it myself.

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it's not quite an identity: if $x=-\pi/2$, then the right-hand side is undefined and the left-hand side is well-defined. – Stefan Smith Dec 27 '13 at 1:24
up vote 8 down vote accepted


What about this approach? $$\frac a b=\frac c d\iff ad=bc$$

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Ooooh, of course! I was so stuck in the pattern of doing the sides separately that I didn't even consider this way. Thanks, this helped a lot, it was exactly what I was looking for! – Threethumb Dec 26 '13 at 21:12
Very easy! If I were you I would exactly post this one. :-) – S. Snape Dec 27 '13 at 8:25

multiply the numerator and denominator of the left side of the by $\sin {x}+ \cos{s}+1$

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cos $x$, presumably. – Eric Stucky Dec 26 '13 at 20:11

For verifying the identity where you already know the simplified answer, Sami's approach is the most direct. Here's what I would do if I didn't already know the right-hand side: whenever I see mixed sines and cosines in a denominator, I want to turn as many of them into squared trig functions as possible, so that I can use the main trig identity $\sin^2+\cos^2=1$. To that end, I would try multiplying top and bottom of your first fraction by $\sin x + \cos x + 1$, and then simplifying.

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Let $t=\tan(x/2)$; then it's well known that $\sin x=2t/(1+t^2)$ and $\cos x=(1-t^2)/(1+t^2)$; therefore, canceling common denominators, $$ \frac{\sin x-\cos x+1}{\sin x+\cos x-1}= \frac{2t-1+t^2+1+t^2}{2t+1-t^2-1-t^2}= \frac{2t(1+t)}{2t(1-t)}=\frac{1+t}{1-t} $$ Can you simplify the other side in a similar way?

Of course, verifying that $$ (\sin x-\cos x+1)\cos x=(\sin x+1)(\sin x+\cos x-1) $$ is easier, but transforming expressions in $\sin x$ and $\cos x$ in the proposed way is a rather powerful tool in other situations.

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Isn't the $t=\tan \frac{x}{2}$ substitution called the Weirstrasse substitution? – K. Rmth Dec 26 '13 at 20:35
@K.Rmth Yes, sometimes. – egreg Dec 26 '13 at 20:38

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