Prove that : `(tantheta)/(1-cottheta)+(cottheta)/(1-tantheta)=1+sectheta" cosec "theta`

Explanation

L.H.S.
`=frac{tan theta}{1-cot theta}+frac{cot theta}{1-tan theta}`

`=frac{frac{sin theta}{cos theta}}{frac{sin theta-cos theta}{sin theta}}+frac{frac{cos theta}{sin theta}}{frac{cos theta-sin theta}{cos theta}}`

`=frac{sin ^{2} theta}{cos theta(sin theta-cos theta)}+frac{cos ^{2} theta}{sin theta(cos theta-sin theta)}`

`=frac{sin ^{2} theta}{cos theta(sin theta-cos theta)}-frac{cos ^{2} theta}{sin theta(sin theta-cos theta)}`

`=frac{sin ^{3} theta-cos ^{3} theta}{sin theta cos theta(sin theta-cos theta)}`

= `{left(sin ^{2} theta+cos ^{2} theta+cos theta sin theta)(sin theta-cos theta)}/{sin theta cos theta(sin theta-cos theta)}`

`=frac{(1+cos theta sin theta)}{sin theta cos theta}`

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