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Question
Prove the following:
(sin α + cos α)(tan α + cot α) = sec α + cosec α
Solution
L.H.S = (sin α + cos α)(tan α + cot α)
= `(sin alpha + cos alpha)(sin alpha/cos alpha + cos alpha/sin alpha)` ...`[∵ tan theta = sin theta/costheta "and" cot theta = cos theta/sin theta]`
= `(sin alpha + cos alpha)((sin^2alpha + cos^2alpha)/(sin alpha * cos alpha))`
= `(sin alpha + cos alpha) * 1/((sin alpha * cos alpha))` ...[∵ sin2θ + cos2θ = 1]
= `1/cosalpha + 1/sinalpha` ...`[∵ sec theta = 1/costheta "and" "cosec" theta = 1/sintheta]`
= sec α + cosec α
= R.H.S
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