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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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