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Question
Prove the following identities:
`(1 + tan^2 "A") + (1 + 1/tan^2"A") = 1/(sin^2 "A" - sin^4"A")`
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Solution
L.H.S. = `(1 + tan^2 "A") + (1 + 1/tan^2"A")`
= (sec2A) + (1 + cot2A)
= sec2A + cosec2A
= `1/cos^2"A" + 1/sin^2"A"`
= `(sin^2"A" + cos^2"A")/(sin^2"A" cos^2"A")`
= `1/(sin^2 "A" cos^2"A")`
= `1/(sin^2"A"(1 - sin^2"A")`
= `1/(sin^2"A" - sin^4"A")`
= R.H.S.
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