How do you verify $cosx/(1-tanx) + sinx/(1-cotx) = sinx + cosx$ ?

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Answer 1 · 2018-03-25T08:06:16

$LHS=cosx/(1-tanx) + sinx/(1-cotx)$

$=cos^2x/(cosx(1-tanx)) + sin^2x/(sinx(1-cotx))$

$=cos^2x/(cosx-sinx) + sin^2x/(sinx-cosx)$

$=cos^2x/(cosx-sinx) - sin^2x/(cosx-sinx)$

$=(cos^2x - sin^2x)/(cosx-sinx)$

$=((cosx+sinx)(cosx-sinx))/(cosx-sinx)$

$= sinx + cosx=RHS$

How do you verify cosx/(1-tanx) + sinx/(1-cotx) = sinx + cosxcosx/(1-tanx) + sinx/(1-cotx) = sinx + cosx?