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

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

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

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

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

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

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

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

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