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Question

If tanθ=2021, then prove that 1sinθ+cosθ1+sinθ+cosθ=37

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Solution

tanθ=2021

As 1+tan2θ=sec2θ

1+20×2021×21=sec2θ

1+400441=sec2θ

841441=sec2θ

secθ=2921

cosθ=2129...............(1)

As 1=cos2θ+sin2θ

1=21×2129×29+sin2θ

1=441841+sin2θ

1441841=sin2θ

400841=sin2θ

2029=sinθ................(2)

1sinθ+cosθ1+sinθ+cosθ............(3)

Substituting equation (1) and (2) in (3),

12029+21291+2029+2129=1+1291+4129=30297029=3070=37

LHS=RHS
Hence proved.

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