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Question
Prove the following identities:
sec2 A . cosec2 A = tan2 A + cot2 A + 2
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Solution
L.H.S. = sec2 A . cosec2 A
= `1/(cos^2A) * 1/(sin^2A)`
= `1/(cos^2A sin^2A)`
= `(sin^2A + cos^2A)/(cos^2A sin^2A)`
= `1/(cos^2A) + 1/(sin^2A)`
= sec2 A + cosec2 A
= 1 + tan2 A + 1 + cot2 A ...(∵ sec2 A = 1 + tan2 A and cosec2 A = 1 + cot2 A)
= tan2 A + cot2 A + 2 = R.H.S.
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sin4A – cos4A = 1 – 2cos2A. For proof of this complete the activity given below.
Activity:
L.H.S. = `square`
= (sin2A + cos2A) `(square)`
= `1 (square)` ...`[sin^2"A" + square = 1]`
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= `square`
= R.H.S.
