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Question

Prove that:
sin2θtanθ+cos2θcotθ+2sinθcosθ=tanθ+cotθ

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Solution

L.H.S
sin2θtanθ+cos2θcotθ+2sinθcosθ

sin2θsinθcosθ+cos2θcosθsinθ+2sinθcosθ

sin3θcosθ+cos3θsinθ+2sinθcosθ

sin4θ+cos4θsinθcosθ+2sinθcosθ

sin4θ+cos4θ+2sin2θcos2θsinθcosθ

(sin2θ+cos2θ)2sinθcosθ

(1)2sinθcosθ

1sinθcosθ

sin2θ+cos2θsinθcosθ

sinθcosθ+cosθsinθ

tanθ+cotθ

R.H.S
Hence, proved.

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