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Prove the identity (sin2x)/(1+(tanx)^2) = 2sinx(cosx)^3

For this question you need to be aware of some of the trigonometric identities, which include the pythagorean identities and the double angle identities.
Since we are aware that sin2x=2sinxcosx and 1 + tan²x= sec²x, we will begin with using these to simplify our fraction.
Step 1: Replace sin2x with its double angle identity
= 2sinxcosx/(1 + tan²x)
Step 2: Replace (1 + tan²x) with sec²x
= 2sinxcosx/sec²x
Step 3: Since secx=1/cosx, so sec²x= 1/ cos²x. Hence you replace sec²x in the equation now
= 2sinxcosx/(1/cos²x)
Step 4: Take the reciprocal of 1/cos²x, which basically involved switching the numerator with the denominator and vice versa
= 2sinxcosx(cos²x)
Step 5: Expand and simplify the equation
= 2sinxcos³x

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