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Sum

Prove the following:

(sinθ + cosecθ)^{2} + (cosθ + secθ)^{2 }= tan^{2}θ + cot^{2}θ + 7

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#### Solution

L.H.S. = (sinθ + cosecθ)^{2} + (cosθ + secθ)^{2 }

= sin^{2}θ + cosec^{2}θ + 2sinθ cosecθ + cos^{2}θ + sec^{2}θ + 2cosθ secθ

= (sin^{2}θ + cos^{2}θ) + cosec^{2}θ + 2 + sec^{2}θ + 2

= 1 + (1 + cot^{2}θ) + 2 + (1 + tan^{2}θ) + 2

= tan^{2}θ + cot^{2}θ + 7

= R.H.S.

Concept: Fundamental Identities

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