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Question
Prove that :(sinθ+cosecθ)2+(cosθ+ secθ)2 = 7 + tan2 θ+cot2 θ.
Solution
L.H.S : (sinθ+cosecθ)2 +(cosθ+secθ)2
=sin2θ + cosec2θ + 2 +cos2θ + sec2θ + 2 `[because sin θ = 1/(cosecθ) " and cos "θ = 1/ (secθ)]`
= sin2θ + cos2θ+1+cot2θ+1+tan2θ+4 `[because cosec^2θ+1+cot^2θ" and "sec^2 θ =1+ tan^2θ]`
=sin2θ+cos2θ+1+cot2θ+1+tan2θ+4 `[because cosec^2θ+1+cot^2θ " and" sec^2 θ=1 +tan^2θ]`
=1+1+1+4+tan2θ+cot2θ `[because cos^2θ+ sin^2θ=1]`
=7+ tan2θ+cot2θ
L.H.S-R.H.S
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