Question

prove that:
( cot theta + cosec theta + 1 ).(cot theta - cosec theta -1).tyan theta = 2

Answer 1

$$\begin{gathered} \mathrm{Your}\mathrm{question}\mathrm{should}\mathrm{be}\mathrm{like}\mathrm{this}: \\ \mathrm{Prove}\mathrm{that}:\left(\cot \mathrm{\theta }+\mathrm{cosec}\mathrm{\theta }+1\right)\left(\cot \mathrm{\theta }-\mathrm{cosec}\mathrm{\theta }+1\right)\tan \mathrm{\theta }=2 \\ \mathrm{Here}\mathrm{is}\mathrm{the}\mathrm{answer}: \\ \mathrm{LHS}=\left(\cot \mathrm{\theta }+\mathrm{cosec}\mathrm{\theta }+1\right)\left(\cot \mathrm{\theta }-\mathrm{cosec}\mathrm{\theta }+1\right)\tan \mathrm{\theta } \\ =\left[\left(\cot \mathrm{\theta }+1\right)+\mathrm{cosec}\mathrm{\theta }\right]\left[\left(\cot \mathrm{\theta }+1\right)-\mathrm{cosec}\mathrm{\theta }\right]\tan \mathrm{\theta } \\ =\left[{\left(\cot \mathrm{\theta }+1\right)}^2-{\mathrm{cosec}}^2\mathrm{\theta }\right].\tan \mathrm{\theta } \\ =\left({\cot }^2\mathrm{\theta }+1+2\cot \mathrm{\theta }-{\mathrm{cosec}}^2\mathrm{\theta }\right).\tan \mathrm{\theta } \\ =\left({\mathrm{cosec}}^2\mathrm{\theta }+2\cot \mathrm{\theta }-{\mathrm{cosec}}^2\mathrm{\theta }\right).\tan \mathrm{\theta } \\ =2\cot \mathrm{\theta }.\tan \mathrm{\theta } \\ =2\times \frac{1}{\tan \mathrm{\theta }}\times \tan \mathrm{\theta } \\ =2 \\ =\mathrm{RHS} \end{gathered}$$