Question

Prove that $\frac{\cos \theta }{(1+sin\theta )}+\frac{(1+\sin \theta )}{cos\theta }=2sec\theta$

Answer 1

Determine the proof of the expression that is $\frac{\cos \theta }{(1+sin\theta )}+\frac{(1+\sin \theta )}{cos\theta }=2sec\theta$

Solve the L.H.S part:

$$\begin{gathered} \frac{\cos \theta }{(1+sin\theta )}+\frac{(1+\sin \theta )}{cos\theta }=\frac{{\cos }^2\theta +{(1+sin\theta )}^2}{cos\theta (1+sin\theta )}\mathbf{\because }\mathbf{}{\mathbf{(}\mathit{a}\mathbf{+}\mathit{b}\mathbf{)}}^{\mathbf{2}}=a^2+b^2+2.a.b \\ \Rightarrow =\frac{{\cos }^2\theta +{\left(1\right)}^2+si{n}^2\theta +2.sin\theta .1}{cos\theta (1+sin\theta )} \\ \Rightarrow =\frac{{\cos }^2\theta +1+si{n}^2\theta +2sin\theta }{cos\theta (1+sin\theta )}\mathbf{\because }\mathbf{}{\mathbf{cos}}^{\mathbf{2}}\mathbf{}\mathit{\theta }\mathbf{+}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathbf{}\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}1 \\ \Rightarrow =\frac{1+1+2sin\theta }{cos\theta (1+sin\theta )} \\ \Rightarrow =\frac{2+2sin\theta }{cos\theta (1+sin\theta )} \\ \Rightarrow =\frac{2}{cos\theta } \\ \Rightarrow =2sec\theta \end{gathered}$$

Hence, the L.H.S = R.H.S.