Question

Prove the following:

(sin α + cos α)(tan α + cot α) = sec α + cosec α

Answer 1

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