Question

prove that cos○/1-tan○+sin/1-cot○=sin○+cos○​

Answer 1

cosx/(1-tanx)+sinx/(1-cotx)

=cosx/(1-tanx)+sinx/(1-1/tanx)

=cosx/(1-tanx)+sinxtanx/(tanx-1)

=cosx/(1-tanx)-sinxtanx/(1-tanx)

=(cosx-sinxtanx)/(1-tanx)

=(cosx-(sinx(sinx/cosx)))/(1-sinx/cosx)

=((cosx)^2-(sinx)^2)/(cosx-sinx)

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

=cosx+sinx

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Answer 2

$\frac{cos \theta}{1 - tan \theta} + \frac{sin \theta}{1 - cot \theta} \\ \\ = \frac{ {cos}^{2} \theta}{cos \theta - sin \theta} + \frac{ {sin}^{2} \theta }{sin \theta - cos \theta} \\ \\ = \frac{ {cos}^{2} \theta}{cos \theta - sin \theta} - \frac{ {sin}^{2} \theta }{cos \theta - sin \theta} \\ \\ = \frac{ {cos}^{2} \theta - {sin}^{2} \theta}{cos \theta - sin \theta} \\ \\ = cos \theta - sin \theta$