Can Trig Identities Be Proven Using Basic Trigonometric Functions?

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ttpp1124
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I believed I've proved it right, but can someone confirm?
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I believe it is correct, but note that

$$(\tan^2 x -1)/\sec^2 x = (\tan^2 x -1) \cos^2 x = \sin^2 x - \cos^2 x = - \cos(2x)$$

so you could have been a little faster.
 
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