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Prove that: ...

Question

Prove that: $\frac{cos A - sin A + 1}{cos A + sin A - 1} = c o sec A + cot A$

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Solution

$L H S = \frac{(cos A - sin A) + 1}{(cos A + sin A) - 1}$

Taking conjugate

$= \frac{(cos A - sin A) + 1}{(cos A + sin A) - 1} \times \frac{(cos A + sin A) + 1}{(cos A + sin A) + 1}$

$= \frac{{cos}^{2} A - {sin}^{2} A + 2 cos A + 1}{2 sin A cos A}$

$= \frac{{cos}^{2} A - {sin}^{2} A + 2 cos A + {sin}^{2} A + {cos}^{2} A}{2 sin A cos A}$

On solving, we get

$= \frac{2 {cos}^{2} A + 2 cos A}{2 sin A cos A}$

$= \frac{cos A + {cos}^{2} A}{sin A cos A}$

$= \frac{cos A}{sin A cos A} + \frac{{cos}^{2} A}{sin A cos A}$

$= cosec A + cot A = R H S$

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