Question

prove that

( sin theta +cos theta ) (cot theta + tan theta) =sec theta +cosec theta.

Answer 1

$$\begin{gathered} \mathrm{LHS}=\left(\sin \mathrm{\theta }+\cos \mathrm{\theta }\right)\left(\cot \mathrm{\theta }+\tan \mathrm{\theta }\right) \\ =\left(\sin \mathrm{\theta }+\cos \mathrm{\theta }\right)\left(\frac{\cos \mathrm{\theta }}{\sin \mathrm{\theta }}+\frac{\sin \mathrm{\theta }}{\cos \mathrm{\theta }}\right) \\ =\left(\sin \mathrm{\theta }+\cos \mathrm{\theta }\right)\left(\frac{{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }}{\sin \mathrm{\theta }\cos \mathrm{\theta }}\right) \\ =\left(\sin \mathrm{\theta }+\cos \mathrm{\theta }\right)\left(\frac{1}{\sin \mathrm{\theta }\cos \mathrm{\theta }}\right)\left[\mathrm{as},{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }=1\right] \\ =\frac{\left(\sin \mathrm{\theta }+\cos \mathrm{\theta }\right)}{\sin \mathrm{\theta }\cos \mathrm{\theta }} \\ =\frac{\sin \mathrm{\theta }}{\sin \mathrm{\theta }\cos \mathrm{\theta }}+\frac{\cos \mathrm{\theta }}{\sin \mathrm{\theta }\cos \mathrm{\theta }} \\ =\frac{1}{\cos \mathrm{\theta }}+\frac{1}{\sin \mathrm{\theta }} \\ =\sec \mathrm{\theta }+\mathrm{cosec}\mathrm{\theta } \\ =\mathrm{RHS} \end{gathered}$$