sinxcosx/cos^2x-sin^2x=tanx/1-tan^2x

Question

Please answer as soon as possible. Thanks!!

Answer 1

This looks like you are trying to "prove" a trigonometric identity. That means you want to work exclusively on one side or the other. Usually when one side of the identity consists of sin/cos and the other side uses other trig functions, you should consider converting everything to sin and cos.

Use the identity that tanx = sinx/cosx to rewrite the right hand side of the identity. Let's just show the changes to the right hand side and see how it can be transformed to look "identical" to the left hand side.

sinx/cosx

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1 - sin²x/cos²x

This is a complex fraction. Multiply numerator and denominator of the "big" fraction by the LCD of all the "little" denominators. That happens to be cos²x.

Doing that immediately gives the desired result because sinx/cosx (cos²x) = sinx cosx, and

(1- sin²x/cos²x) (cos²x) = cos²x - sin²x