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Question
Prove the following identities:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
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Solution
L.H.S. = (sin A + cosec A)2 + (cos A + sec A)2
= sin2 A + cosec2 A + 2 sin A cosec A + cos2 A + sec2 A + 2 cos A sec A
= `sin^2A + cosec^2A + 2sinA xx 1/sinA + cos^2A + sec^2A + 2cosA xx 1/cosA`
= sin2 A + cos2 A + cosec2 A + sec2 A + 2 + 2 ...(∵ sin2 A + cos2 A = 1)
= 1 + cosec2 A + sec2 A + 4
= (1 + cot2 A) + (1 + tan2 A) + 5 ...[∵ cosec2 A = 1 + cot2 A and sec2 A = 1 + tan2 A]
= 1 + cot2 A + 1 + tan2 A + 5
= 7 + tan2 A + cot2 A
= R.H.S.
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Complete the following activity to prove:
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Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
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= `1/(sinθ xx cosθ)` ....... ∵ `square`
= `1/sinθ xx 1/cosθ`
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∴ L.H.S. = R.H.S.
