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Question
Prove that `sqrt((1 + cos "A")/(1 - cos"A"))` = cosec A + cot A
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Solution
L.H.S = `sqrt((1 + cos "A")/(1 - cos"A"))`
= `sqrt((1 + cos "A")/(1 - cos "A") xx (1 + cos "A")/(1 + cos "A"))` ......[On rationalising the denominator]
= `sqrt((1 + cos "A")^2/(1 - cos^2 "A"))`
= `sqrt((1 + cos "A")^2/(sin^2 "A")` ......`[(because sin^2"A" + cos^2"A" = 1),(therefore 1 - cos^2"A" = sin^2"A")]`
= `(1 + cos"A")/"sin A"`
= `1/"sin A" + "cos A"/"sin A"`
= cosec A + cot A
= R.H.S
∴ `sqrt((1 + cos "A")/(1 - cos"A"))` = cosec A + cot A
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