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Question

Prove that cotθcscθ+1+cscθ+1cotθ=2secθ

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Solution

cotθcscθ+1+cscθ+1cotθ=2secθ
Taking L.H.S., we have
cotθcscθ+1+cscθ+1cotθ
=cot2θ+(cscθ+1)2(cscθ+1)cotθ
=cot2θ+(csc2θ+1+2cscθ)(cscθ+1)cotθ
=cot2θ+1+csc2θ+2cscθ(cscθ+1)cotθ
=csc2θ+csc2θ+2cscθ(cscθ+1)cotθ(1+cot2θ=csc2θ)
=2csc2θ+2cscθ(cscθ+1)cotθ
=2cscθ(cscθ+1)(cscθ+1)cotθ
=2cscθcotθ
=2(1sinθ)(cosθsinθ)
=2cosθ
=2secθ
=R.H.S.
Hence, it is proved that L.H.S. = R.H.S., i.e., 2secθ.

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