Question

Prove that $\left(\frac{\cos A}{(1+\sin A)}\right)+\left(\frac{(1+\sin A)}{\cos A}\right)=2secA$

Answer 1

Determine the proof of the expression that is $\left(\frac{\cos A}{(1+\sin A)}\right)+\left(\frac{(1+\sin A)}{\cos A}\right)=2secA$

Solve the L.H.S part:

$$\begin{gathered} \left(\frac{\cos A}{(1+\sin A)}\right)+\left(\frac{(1+\sin A)}{\cos A}\right)=\frac{{\cos }^{2}A+{(1+\sin A)}^2}{\cos A(1+\sin A)}\because \mathbf{}{\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}}^{\mathbf{2}}\mathbf{}=a^2+b^2+2.a.b \\ \Rightarrow =\frac{{\cos }^{2}A+{\left(1\right)}^2+{\sin }^{2}A+2.\sin A.1}{\cos A(1+\sin A)} \\ \Rightarrow =\frac{{\cos }^{2}A+1+{\sin }^{2}A+2\sin A}{\cos A(1+\sin A)}\mathbf{\because }\mathbf{}{\mathbf{cos}}^{\mathbf{2}}\mathbf{}\mathit{A}\mathbf{+}\mathbf{}{\mathbf{sin}}^{\mathbf{2}}\mathbf{}\mathit{A}\mathbf{}\mathbf{=}1 \\ \Rightarrow =\frac{1+1+2\sin A}{\cos A(1+\sin A)} \\ \Rightarrow =\frac{2+2\sin A}{\cos A(1+\sin A)} \\ \Rightarrow =\frac{2(1+\sin A)}{\cos A(1+\sin A)} \\ \Rightarrow =\frac{2}{\cos A} \\ \Rightarrow =2secA \end{gathered}$$

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