The problem is to simplify the expression $D = \sqrt{\frac{2}{1+sin\alpha} + \frac{2}{1-sin\alpha}}$.

1. Problem Description

The problem is to simplify the expression $D = \sqrt{\frac{2}{1+sin\alpha} + \frac{2}{1-sin\alpha}}$.

2. Solution Steps

First, combine the two fractions inside the square root by finding a common denominator.

$D = \sqrt{\frac{2(1-sin\alpha) + 2(1+sin\alpha)}{(1+sin\alpha)(1-sin\alpha)}}$

$D = \sqrt{\frac{2 - 2sin\alpha + 2 + 2sin\alpha}{1 - sin^2\alpha}}$

$D = \sqrt{\frac{4}{1 - sin^2\alpha}}$

We know the trigonometric identity:

$sin^2\alpha + cos^2\alpha = 1$

$cos^2\alpha = 1 - sin^2\alpha$

Substitute $cos^2\alpha$ for $1 - sin^2\alpha$:

$D = \sqrt{\frac{4}{cos^2\alpha}}$

$D = \frac{\sqrt{4}}{\sqrt{cos^2\alpha}}$

$D = \frac{2}{|cos\alpha|}$

3. Final Answer

$D = \frac{2}{|cos\alpha|}$