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Question
Prove the following identities:
(sinθ + sec θ)2 + (cosθ + cosec θ)2 = (1 + cosecθ sec θ)2
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Solution
L.H.S. = (sinθ + sec θ)2 + (cosθ + cosec θ)2
= `(sin theta + 1/cos theta)^2 + (cos theta + 1/sin theta)^2`
= `(sin theta cos theta + 1)^2/cos^2 theta + (sin theta cos theta + 1)^2/sin^2 theta `
= `(sin theta cos theta + 1)^2 (1/cos^2theta + 1/sin^2 theta)`
= `(sin theta cos theta + 1)^2 ((sin^2 theta + cos^2 theta)/(sin^2 theta cos^2 theta))`
= `(sin theta cos theta + 1)^2 (1/(sin^2 theta cos^2 theta))`
= `((sin theta cos theta + 1)/(sin theta cos theta))^2`
= `((sin theta cos theta)/(sin theta cos theta) + 1/(sin theta cos theta))^2`
= (1 + cosecθ secθ)2
= R.H.S.
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