Question

prove that tan theta/(1 - cos theta) + tan theta/(1+ cos theta) = 2sec theta .cosec theta

Answer 1

$$\begin{gathered} \mathrm{Taking}\mathrm{LHS}, \\ \frac{\mathrm{tan\theta }}{1-\mathrm{cos\theta }}+\frac{\mathrm{tan\theta }}{1+\mathrm{cos\theta }} \\ =\mathrm{tan\theta }\left[\frac{1+\mathrm{cos\theta }+1-\mathrm{cos\theta }}{1-{\cos }^2\mathrm{\theta }}\right] \\ =\frac{2\mathrm{tan\theta }}{{\sin }^2\mathrm{\theta }}=2\times \frac{\mathrm{sin\theta }}{\mathrm{cos\theta }}\times \frac{1}{{\sin }^2\mathrm{\theta }} \\ =2\mathrm{sec\theta cosec\theta } \\ =\mathrm{RHS} \end{gathered}$$