Expand the term (1 + sin A)² using the algebraic identity (a + b)² = a² + 2ab + b². Then, group cos² A and sin² A together.

Using the trigonometric identity cos² A + sin² A = 1, we replace the grouped terms with 1. Adding 1 and 1 gives 2 + 2sin A in the numerator.

Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a

Answer: $\text{LHS} = \frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2\sec A = \text{RHS}$ Hence proved.

Step-by-step solution

Step 1: Take the Left Hand Side (LHS) and find the common denominator

We begin by considering the left hand side of the equation. To add the 2 fractions, we find the common denominator, which is $(1 + \sin A)\cos A$ , and combine the numerators.

Step 2: Expand the numerator

Expand the term $(1 + \sin A)^2$ using the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$ . Then, group $\cos^2 A$ and $\sin^2 A$ together.

Step 3: Apply the identity $\cos^2 A + \sin^2 A = 1$

Using the trigonometric identity $\cos^2 A + \sin^2 A = 1$ , we replace the grouped terms with 1. Adding 1 and 1 gives $2 + 2\sin A$ in the numerator.

Step 4: Factor out the common factor and simplify to the RHS

We factor out 2 from the numerator to get $2(1 + \sin A)$ . Cancelling the common factor $(1 + \sin A)$ from the numerator and denominator leaves $\frac{2}{\cos A}$ , which equals $2\sec A$ since $\sec A = \frac{1}{\cos A}$ .

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