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Question

If cosecθ+cotθ=k then prove that cosθ=k21k2+1

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Solution

cosecθ+cotθ=k

1sinθ+cosθsinθ=k

k2=(1+cosθ)2sin2θ

k21k2+1=(1+cosθ)2sin2θ1(1+cosθ)2sinθ+1

=1+cos2+2cosθsin2θ1+cos2+2cosθ+sin2θ

sin2θ+cos2θ+cos2θ+2cosθsin2θ1+(cos2θ+sin2θ)+2cosθ

2cos2θ+2cosθ2+2cosθ(sin2θ+cos2θ=1)

=cosθ(2+2cosθ)(2+2cosθ)

=cosθ

cosθ=k21k2+1
Hence proved

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