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Question
Prove the following identities:
`cottheta/("cosec" theta - 1) = ("cosec" theta + 1)/cot theta`
Solution
L.H.S. = `cottheta/("cosec" theta - 1)`
= `cottheta/("cosec" theta - 1) xx ("cosec" theta + 1)/("cosec" theta + 1)`
= `(cottheta("cosec" + 1))/("cosec"^2theta - 1)`
= `(cot theta("cosec" theta + 1))/cot^2 theta ...[(because 1 + cot^2theta = "cosec"^2theta),(therefore "cosec"^2theta - 1 = cot^2theta)]`
= `("cosec" theta + 1)/cot theta`
= R.H.S.
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