Prove the following identity: `(1+cottheta-cos e ctheta)(1+t a ntheta+s e ctheta)=2`

Explanation

`L.H.S` `=(1+cot theta-operatorname{cosec} theta)(1+tan theta+sec theta)`


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


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


`=frac{1}{sin theta cos theta}(csin theta cos theta+sin ^{2} theta+sin theta+cos ^{2} theta +sin theta cos theta+cos theta-cos theta-sin theta-1)`


`=frac{1}{sin theta cos theta}(2 sin theta cos theta+(sin ^{2} theta+cos ^{2} theta)-1)`


`=frac{1}{sin theta cos theta}(2 sin theta cos theta+1-1)`


`=frac{2 sin theta cos theta}{sin theta cos theta}`


`=2`


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