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

Explanation

LHS = `(cosec theta-sin theta)(sec theta-cos theta)(tan theta+cot theta)`


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


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


`= frac{cos ^{2} theta}{sin theta} times frac{sin ^{2} theta}{cos theta} times frac{1}{sin theta cdot cos theta}`


`= 1`


= RHS