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how to tell that what trigonometric identity (a.k.a. Pythagorean trigonometric identity) will work on the given equation , so then you can simplify the equation accordingly in order to apply that identity?

I'm talking about applying these if you still dont know what i'm saying :

$\sin^2A + \cos^2 A = 1,$

$\tan^2 A + 1 = \sec^2 A,$

$\cot^2 A + 1= \csc^2 A$

Examples would be great, thanks in advance.

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Not sure what you're asking. All 3 equations use different trigonometric functions, yet they are all basically the same identity. Take the first and divide by $\cos^2A$ to get the second or $\sin^2A$ to get the third. – Mike Sep 16 '12 at 19:52
The way to tell what will work is by trying things to see what works. Do enough of that, and you will develop a sense of what works. In other words, the way to do this is for you to do examples, not for others to do examples for you. – Gerry Myerson Sep 17 '12 at 6:26

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