Question

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

Answer 1

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

L.H.S $=\frac{\text{cos A}}{(1 + \text{sin A})} + \frac{(1 +\text{ sin A})}{\text{cos A }}$

$= \frac{[\text{cos² A} + (1 +\text{ sin A)²}] }{ [(1 + \text{sin A})(\text{cos A})]}$

$=\frac{ (\text{cos² A + 1 + sin² A + 2sin A}) }{ (\text{1 + sin A})(\text{cos A})}$

$=\frac{ [\text{sin² A + cos² A + 1 + 2sin A}] }{ [\text{(1 + sin A)(cos A})]}$

$= \frac{(1 + 1 + \text{2sin A}) }{ [(1 + \text{sin A})(\text{cos A})]}$

$= \frac{(2 + \text{2sin A}) }{ [(1 + \text{sin A})(\text{cos A})]}$

$= \frac{(2 +\text{ 2sin A}) }{ [(\text{1 + sin A)(cos A)}]}$

$=\frac{\text{ 2(1 + sin A) }}{ [\text{(1 + sin A)(cos A)}]}$

$=\frac{ 2}{\text{cos A}}$
$=\text{ 2sec A}$
= R.H.S