Question

prove that tan theta/1-cot theta +cot theta/1-tan theta=1+tan theta+cot theta



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Answer 1

$LHS=\frac{tan{\theta}}{1-cos{\theta}}+\frac{cot{\theta}}{1-tan{\theta}}\\\\=\frac{tan{\theta}}{1-cot{ \theta}}+\frac{cot{\theta}}{1-\frac{1}{cot{\theta}}}$
$=\frac{tan{\theta}}{1-cot{\theta}}-\frac{cot^2{\theta}}{1-cot{\theta}}$
$=\frac{\frac{1}{cot{\theta}}-cot^2{\theta}}{1-cot{\theta}}$
$=\frac{1-cot^3{\theta}}{cot{\theta}(1-cot{\theta})}$
$=\frac{(1-cot{\theta})(1+cot^2{ \theta}+cot{\theta})}{cot{\theta}(1-cot{\theta})}$
$=\frac{1}{cot{\theta}}+\frac{cot^2{\theta}}{cot{\theta}}+\frac{cot{\theta}}{cot{\theta}}$
$= tan{\theta}+cot{\theta}+1=RHS$

Answer 2

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