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Question
Prove that:
`tanA/(1 - cotA) + cotA/(1 - tanA) = secA "cosec" A + 1`
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Solution
L.H.S. = `tanA/(1 - cotA) + cotA/(1 - tanA)`
= `tanA/(1 - 1/tanA) + (1/tanA)/(1 - tanA)`
= `tan^2A/(tanA - 1) + 1/(tanA(1 - tanA))`
= `(tan^3A - 1)/(tanA(1 - tanA))`
= `((tanA - 1)(tan^2A + 1 + tanA))/(tanA(tanA - 1)`
= `(sec^2A + tanA)/tanA`
= `(1/cos^2A)/(sinA/cosA) + 1`
= `1/(sinAcosA) + 1`
= sec A cosec A + 1 = R.H.S.
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