Question

Prove that `\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin\theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2\sin^{2}\theta -1}`

Answer 1

`LHS=\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta}+\frac{\sin \theta +\cos \theta }{\sin \theta -\cos \theta }`

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

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

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

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