Q8 of 104 Page 204

Prove the following by using trigonometric identities:

(sinθ + cosecθ)2 + (cosθ + secθ)2 = 7 + tan2θ + cot2θ.

L.H.S

(sinθ + cosecθ)2 + (cosθ + secθ)2


= sin2θ + 2sinθcosecθ + cosec2θ + cos2θ + 2cosθsecθ + sec2θ


= sin2θ + 2 + cosec2θ + cos2θ + 2 + sec2θ


= sin2θ + cos2θ + sec2θ + cosec2θ + 2 + 2


= 1 + 2 + 2 + (1 + tan2θ) + (1 + cot2θ)


= 7 + tan2θ + cot2θ


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