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Question
Prove that:
cos A (1 + cot A) + sin A (1 + tan A) = sec A + cosec A
Solution
L.H.S. = cos A (1 + cot A) + sin A (1 + tan A)
= `cosA(1 + cosA/sinA) + sinA(1 + sinA/cosA)`
= `(cosA(sinA + cosA))/sinA + (sinA(cosA + sinA))/cosA`
= `(sinA + cosA)[cosA/sinA + sinA/cosA]`
= `(sinA + cosA)[(cos^2A + sin^2A)/(sinAcosA)]`
= `(sinA + cosA) xx 1/(sinAcosA)`
= `(sinA + cosA)/(sinAcosA)` ...[∵ cos2θ + sin2θ = 1]
= `sinA/(sinAcosA) + cosA/(sinAcosA)`
= `1/cosA + 1/sinA`
= sec A + cosec A = R.H.S.
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