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Question

Identity transformations of Trigonometric Expressions.
prove the following identities.
sinx+cosxsinxcosxsec2x+2tan2x1=2tanx+1.

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Solution

LHS=sinx+cosxsinxcosxsec2x+2tan2x1
= multiply & divide by
(sinx+cosx)
=(sinx+cosx)2cos2x(2cos2x+1sin2xcos2x)
=1+sin2xcos2x2cos2x+1(cos2x)
=1+sin2x2cos2x1(cos2x)
=2sinxcosx2cos2x(cos2x)
=2cosx(cos2x)[sinxcosx]
=2cosxsin2xcos2x[sinxcosx]
=2cosxsinx+cosx=(2tanx+1)
=RHS
Hence proved!

1127719_887009_ans_1425f78ee4d34555804ffab651c2d621.jpg

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