Prove the following identities: `(cos e ctheta-sintheta)(sectheta-costheta)=1/(tantheta+cottheta)`

Explanation

LHS = `({cosec} theta-sin theta)(sec theta-cos theta)`


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


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


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


`=sin theta cos theta` …….(1)


RHS = `frac{1}{tan theta+cot theta}`


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


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


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