Prove the following identities: `(sintheta+sectheta)^2+(costheta+cos e ctheta)^2=(1+secthetacos e ctheta)^2`

Explanation

LHS `=(sin theta+sec theta)^{2}+(cos theta+{cosec} theta)^2 `


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


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


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


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


=(1+sec theta cosec theta)^2= RHS.`