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Question
Prove the following identities:
`((1 + tan^2A)cotA)/(cosec^2A) = tan A`
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Solution
L.H.S. = `((1 + tan^2A)cotA)/(cosec^2A)`
= `(sec^2A cotA)/(cosec^2A` ...(∵ sec2 A = 1 + tan2 A)
= `(1/(cos^2A) xx (cosA)/(sinA))/(1/(sin^2A))`
= `(1/(cosA sinA))/(1/(sin^2A))`
= `sinA/cosA`
= tan A = R.H.S.
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Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
