Question

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$\frac{coseca}{coseca - 1} + \frac{coseca}{coseca + 1} = 2 {sec}^{2} a$
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Answer 1

Solution

To prove:

$\sf{ \frac{cosec \: x}{cosec \: x - 1} + \frac{cosec \: x}{cosec \: x + 1} = 2sec {}^{2}x }$

LHS:

$\sf{ \frac{cosec \: x(cosec \: x + 1) + cosec \: x(cosec \: x - 1)}{(cosec \: x + 1)(cosec \: x - 1)} } \\ \\ = \sf{\frac{cosec {}^{2}x + \cancel{cosec \: x} + cosec {}^{2}x - \cancel{cosec \: x} }{cosec {}^{2}x - 1 }} \\ \\ = \sf{\frac{2cosec{}^{2}x }{cot{}^{2}x }} \\ \\ = \sf{\frac{ \frac{2}{ \cancel{sin {}^{2}x}} }{ \frac{cos {}^{2}x }{ \cancel{sin {}^{2}x }}}} \\ \\ = \sf{\frac{2}{cos{}^{2}x }} \\ \\ = \sf{2sec{}^{2}x }$

Hence,proved

Answer 2

Answer:

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