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Question
Prove the following identity :
`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`
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Solution 1
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Solution 2
LHS = `(1 + tan^2A) + (1 + 1/tan^2A)`
= `(1 + sin^2A/cos^2A) + (1 + 1/(sin^2A/cos^2A))`
= `((cos^2A + sin^2A)/(cos^2A)) + ((cos^2A + sin^2A)/(sin^2A))`
= `1/(1 - sin^2A) + 1/sin^2A` (∵ `cos^2A + sin^2A = 1`)
= `(sin^2A + 1 - sin^2A)/(sin^2A(1 - sin^2A)) = 1/(sin^2A - sin^4A)`
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