Question 5 (ii)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(ii) $\frac{c o s A}{(1 + s i n A)} + \frac{(1 + s i n A)}{c o s A} = 2 s e c A$

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Solution

(ii) $\frac{c o s A}{(1 + s i n A)} + \frac{(1 + s i n A)}{c o s A} = 2 s e c A$
$L . H . S . = \frac{c o s A}{(1 + s i n A)} + \frac{(1 + s i n A)}{c o s A} = \frac{[c o s^{2} A + (1 + s i n A)^{2}]}{(1 + s i n A) c o s A} = \frac{(c o s^{2} A + s i n^{2} A + 1 + 2 s i n A)}{(1 + s i n A) c o s A} = \frac{(1 + 1 + 2 s i n A)}{(1 + s i n A) c o s A} = \frac{(2 + 2 s i n A)}{(1 + s i n A) c o s A} = \frac{2 (1 + s i n A)}{(1 + s i n A) c o s A} = \frac{2}{c o s A} = 2 s e c A = R . H . S .$

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