Question

Evaluate : Root Cosec square theta + Sec square theta

Answer 1

$$\begin{gathered} \sqrt{{\mathrm{cosec}}^2\mathrm{\theta }+{\sec }^2\mathrm{\theta }} \\ =\sqrt{1+{\cot }^2\mathrm{\theta }+1+{\tan }^2\mathrm{\theta }} \\ =\sqrt{{\tan }^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }+2} \\ =\sqrt{{\left(\tan \mathrm{\theta }\right)}^2+{\left(\cot \mathrm{\theta }\right)}^2+2\times \tan \mathrm{\theta }\times \cot \mathrm{\theta }}\left[\mathbf{as}\mathbf{,}\mathbf{}\mathbf{tan}\mathbf{}\mathbf{\theta }\mathbf{}\mathbf{.}\mathbf{}\mathbf{cot}\mathbf{}\mathbf{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\right] \\ =\sqrt{{\left(\tan \mathrm{\theta }+\cot \mathrm{\theta }\right)}^2} \\ =\left(\tan \mathrm{\theta }+\cot \mathrm{\theta }\right) \end{gathered}$$