Question

Prove that
$\sin \theta (1+\tan \theta )+cos\theta (1+\cot \theta )=\csc \theta +\sec \theta$

Answer 1

$\begin{array}{c}L.H.S.=\sin \theta \left(1+\tan \theta \right)+\cos \theta \left(1+\cot \theta \right)\\ \Rightarrow \sin \theta \left(1+\frac{\sin \theta }{\cos \theta }\right)+\cos \theta \left(1+\frac{\cos \theta }{\sin \theta }\right)\\ \Rightarrow \sin \theta +\frac{{\sin }^2\theta }{\cos \theta }+\cos \theta +\frac{{\cos }^2\theta }{\sin \theta }\\ \Rightarrow \sin \theta +\frac{{\cos }^2\theta }{\sin \theta }+\cos \theta +\frac{{\sin }^2\theta }{\cos \theta }\\ \Rightarrow \frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta }+\frac{{\cos }^2\theta +{\sin }^2\theta }{\cos \theta }\\ \Rightarrow \frac{1}{\sin \theta }+\frac{1}{\cos \theta }=\cos ec\theta +\sec \theta =R.H.SHenceproved\end{array}$