Question

Prove that ✓(cosec theta +1\cosec theta -1)-✓(cosec theta-1\cosec theta+1)=2cot theta

Answer 1

Correct Question:–

Prove that √(cosec theta +1\cosec theta -1)-√(cosec theta-1\cosec theta+1)=2 tan theta

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To Proof:–

${ \tt{ \sqrt{ \frac{cosec \theta + 1}{cosec \theta - 1} } - \sqrt{ \frac{cosec \theta - 1}{cosec \theta + 1} } = 2tan \theta} }$

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Proof:–

Taking LHS and rationalising the denominator;

$\longrightarrow{ \tt{\sqrt{ \frac{cosec \theta + 1}{cosec \theta - 1} \times \frac{cosec \theta + 1}{cosec \theta + 1} } - \sqrt{ \frac{cosec \theta - 1}{cosec \theta + 1} \times \frac{cosec \theta - 1}{cosec \theta - 1}}}} \\ \\ \\ \longrightarrow \: { \tt{\sqrt{ \frac{(cosec \theta + 1) {}^{2} }{cosec {}^{2} \theta - 1 {}^{2} } } - \sqrt{ \frac{(cosec \theta - 1) {}^{2} }{cosec {}^{2} \theta - 1 {}^{2} } } }} \\ \\ \\ \longrightarrow { \tt{ \frac{cosec \theta + 1}{cot \theta } - \frac{cosec \theta - 1}{cot \theta } }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:{ \sf{(1 + cot {}^{2} \theta = cosec {}^{2} \theta)}} \\ \\ \\ \longrightarrow { \tt{\frac{ \cancel{cosec \theta} + 1 -{ \cancel{ cosec \theta}} + 1}{cot \theta}}} \\ \\ \\ { \tt{\longrightarrow \: \frac{2}{cot \theta} \leadsto \: 2tan \theta}}$

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