Question

If y=(1-tanx/1+tanx) show that dy/dx=-2/1+sin2x

Answer 1

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SOLUTION:--

We have:y=1−tanx1+tanx=1−sinxcosx1+sinxcosx⇒y=cosx−sinxcosx+sinxDifferentiating both sides with respect to x, we get:dydx=(cosx+sinx)ddx(cosx−sinx)−(cosx−sinx)ddx(cosx+sinx)(cosx+sinx)2=(cosx+sinx)ddx(cosx−sinx)−(cosx−sinx)ddx(cosx+sinx)cos2x+sin2x+1sinx.cosx=(cosx+sinx)(−sinx−cosx)−(cosx−sinx)(−sinx+cosx)cos2x+sin2x+2sinx.cosx=−(cosx+sinx)2−(cosx−sinx)21+sin(2x)=−cos2x−sin2x−2sinx.cosx−cos2x−sin2x+2sinx.cosx1+sin(2x)=−2cos2x−2sin2x1+sin(2x)=−2(cos2x+sin2x)1+sin(2x)=−21+sin(2x)⇒dydx=−21+sin(2x)
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