Prove the following identities: `(tantheta-cottheta)/(sinthetacostheta)=sec^2theta-cos e c^2theta=tan^2theta-cot^2theta`

Explanation

LHS `=frac{tan theta-cot theta}{sin theta cos theta}=frac{frac{sin theta}{cos theta}-frac{cos theta}{sin theta}}{sin theta cos theta}=frac{frac{sin ^{2} theta-cos ^{2} theta}{sin theta cos theta}}{cos theta sin theta}`


`Rightarrow frac{sin ^{2} theta-cos ^{2} theta}{sin ^{2} theta cos ^{2} theta}`


`Rightarrow frac{sin ^{2} theta}{sin ^{2} theta cos ^{2} theta}-frac{cos ^{2} theta}{sin ^{2} theta cos ^{2} theta}`


`Rightarrow frac{1}{cos ^{2} theta}-frac{1}{sin ^{2} theta}`


`Rightarrow sec ^{2} theta-cosec^{2} theta`


`=(1+tan ^{2} theta)-(1+cot ^{2} theta)`


`=tan ^{2} theta-cot ^{2} theta`


= RHS.