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Question
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Solution
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
L.H.S = (sin A + cosec A)2 + (cos A + sec A)2
= sin2A + cosec2A + 2sinA cosecA + cos2A + sec2A + 2cosA secA
= (sin2A + cos2A) + (cosec2A + sec2A) + 2sinA`(1/sinA)`+ 2cosA`(1/cosA)`
= (1) + (1 + cot2A + 1 + tan2A) + (2) + (2)
= 7 + tan2A + cot2A
= R.H.S
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