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Question

Prove the following:
1cosecΘcotΘ1sinΘ=1sinΘ1cosecΘ+cotΘ

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Solution

Let the L.H.S of the given equation

LHS=1cosecθcotθ1sinθ=1cosecθcotθ×cosecθ+cotθcosecθ+cotθ1sinθ=cosecθ+cotθcosec2θcot2θ1sinθ=cosecθ+cotθ11sinθ=sinθ(cosecθ+cotθ)1sinθ=1+cosθ1sinθ=1sinθ+cosθsinθ1sinθ=1sinθ+cotθcosecθ=1sinθ(cosecθcotθ)=1sinθ(cosecθcotθ)×(cosecθ+cotθ)(cosecθ+cotθ)=1sinθ1(cosecθ+cotθ)

LHS=RHS

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