Question

prove that cot theta/cosec theta+1=cosec theta +1/cot theta.

Answer 1

$$\begin{gathered} \mathrm{LHS}=\frac{\cot \mathrm{\theta }}{\mathrm{cosec}\mathrm{\theta }+1} \\ =\frac{\cot \mathrm{\theta }}{\mathrm{cosec}\mathrm{\theta }+1}\times \frac{\mathrm{cosec}\mathrm{\theta }-1}{\mathrm{cosec}\mathrm{\theta }-1} \\ =\frac{\cot \mathrm{\theta }\left(\mathrm{cosec}\mathrm{\theta }-1\right)}{{\mathrm{cosec}}^2\mathrm{\theta }-1} \\ =\frac{\cot \mathrm{\theta }\left(\mathrm{cosec}\mathrm{\theta }-1\right)}{{\cot }^2\mathrm{\theta }} \\ =\frac{\mathrm{cosec}\mathrm{\theta }-1}{\cot \mathrm{\theta }} \\ =\mathrm{RHS} \end{gathered}$$