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To prove the given identity:

\[
\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} \equiv 2 \sec x
\]

We will simplify the left-hand side (LHS) step by step and show that it equals the right-hand side (RHS), which is \( 2 \sec x \). Here's the proof process:

### Step 1: Combine the two terms on the LHS

The LHS is:
\[
\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}
\]

To combine these two terms, we need to get a common denominator. The common denominator is:
\[
(1 + \sin x) \cdot \cos x
\]

Now, rewrite each term with this common denominator:

\[
\frac{\cos^2 x}{(1 + \sin x)\cos x} + \frac{(1 + \sin x)^2}{(1 + \sin x)\cos x}
\]

### Step 2: Simplify the numerator

Now that we have a common denominator, we can combine the numerators:

\[
\frac{\cos^2 x + (1 + \sin x)^2}{(1 + \sin x)\cos x}
\]

Let's expand the second term in the numerator:
\[
(1 + \sin x)^2 = 1 + 2\sin x + \sin^2 x
\]

Thus, the numerator becomes:
\[
\cos^2 x + 1 + 2\sin x + \sin^2 x
\]

### Step 3: Use the Pythagorean identity

Recall the Pythagorean identity:
\[
\cos^2 x + \sin^2 x = 1
\]

So, substitute this into the expression:
\[
\cos^2 x + 1 + 2\sin x + \sin^2 x = (1 + \sin^2 x + \cos^2 x) = (identer Add

Prove that (cos x) / (1 + sin x) + (1 + sin x) / (cos x) ≡ 2 sec x for x ≠ (2n+1)π/2

This problem provides a step-by-step solution to proving the trigonometric identity: (cos x) / (1 + sin x) + (1 + sin x) / (cos x) = 2 sec x. The solution involves simplifying expressions using common denominators and Pythagorean identities. Suitable for high school students in Grades 11-12.

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