Question

Prove that tan theta / 1 -cot theta + cot theta / 1 - sin theta = 1 + tan theta + cot theta

Answer 1

Answer:

your answer attached in the photo

Answer 2

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$\red{ \frac{tan \theta}{1- cot \theta } + \frac{cot \theta}{1-tan\theta} = 1+tan \theta + cot \theta }$

$LHS = \frac{tan \theta}{1- cot \theta } + \frac{cot \theta}{1-tan\theta}$

$= \frac{tan \theta}{1- \frac{1}{tan \theta }} + \frac{\frac{1}{tan \theta}}{1-tan\theta}$

$= \frac{ tan^{2} \theta }{ tan \theta - 1 } - \frac{ 1}{tan \theta ( tan \theta - 1 ) }$

$= \frac{ tan^{3} \theta - 1 }{ tan \theta( tan \theta - 1 ) }$

$= \frac{ (tan \theta - 1)( tan^{2} \theta + tan \theta + 1 )}{ tan \theta ( tan \theta - 1 ) }$

$= \frac{( tan^{2} \theta + tan \theta + 1 )}{ tan \theta }$

$= \frac{ tan^{2} \theta}{tan \theta} + \frac{tan \theta}{tan \theta } + \frac{1}{tan \theta }$

$= tan\theta + 1 + cot \theta$

$= RHS$

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