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Question
Prove that `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")` = – 1
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Solution
L.H.S = `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")`
= `"cot A"/(1 - 1/(tan"A")) + "tan A"/(1 - tan "A")`
= `"cot A"/((tan "A" - 1)/(tan "A")) + "tan A"/(1 - tan "A")`
= `"cot A tan A"/(tan "A" - 1) + "tan A"/(1 - tan "A")`
= `1/(tan "A" - 1) + "tan A"/(1 - tan "A")` ......[∵ cot A tan A = 1]
= `- 1/(1 - tan "A") + "tan A"/(1 - tan "A")`
= `- (1/(1 -tan "A") - "tan A"/(1- tan "A"))`
= `-((1 - tan "A")/(1 - tan "A"))`
= – 1
= R.H.S
∴ `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")` = – 1
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