$\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1} = cosec A + cot A$

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\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1} = cosec A + cot A

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L H S = \frac{cos A - sin A + 1}{cos A + sin A - 1} = \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 A + sin A ) ( cos A - sin A ) + ( cos A + sin A ) + ( cos A - sin A ) + 1}{( cos A + sin A ) ^{2} - 1}

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

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

= cot A + cosec A = R H S

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Question Text	$\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1} = cosec A + cot A$
Answer Type	Text solution:1
Upvotes	150