Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$ \frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A $

Given:

\( \frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A \)

To do:

We have to prove the given identity.

Solution:  

LHS $=\frac{\cos \mathrm{A}}{1+\sin \mathrm{A}}+\frac{1+\sin \mathrm{A}}{\cos \mathrm{A}}$

$=\frac{(\cos \mathrm{A})^{2}+(1+\sin \mathrm{A})^{2}}{\cos \mathrm{A}(1+\sin \mathrm{A})}$

$=\frac{\cos ^{2} \mathrm{~A}+1+\sin ^{2} \mathrm{~A}+2 \sin \mathrm{A}}{\cos \mathrm{A}(1+\sin \mathrm{A})}$

$=\frac{1+1+2 \sin \mathrm{A}}{\cos \mathrm{A}(1+\sin \mathrm{A})}$

$=\frac{2+2 \sin \mathrm{A}}{\cos \mathrm{A}(1+\sin \mathrm{A})}$

$=\frac{2(1+\sin \mathrm{A})}{\cos \mathrm{A}(1+\sin \mathrm{A})}$

$=\frac{2}{\cos \mathrm{A}}$

$=2 \sec \mathrm{A}$

$=$ RHS

Hence proved.

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Updated on: 10-Oct-2022

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