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To prove the identity

\[
\cos x + \sin x \tan 2x = \frac{\cos x}{\cos 2x}
\]

we'll work with both sides of the equation and show that they are equivalent.

### Step-by-Step Proof:

1. **Start with the left side:**

   \[
   \cos x + \sin x \tan 2x
   \]

2. **Express \( \tan 2x \) using the double-angle formula:**

   Recall that \( \tan 2x = \frac{2 \sin x \cos x}{\cos^2 x - \sin^2 x} \).

   Substitute this into the expression:

   \[
   \cos x + \sin x \cdot \frac{2 \sin x \cos x}{\cos^2 x - \sin^2 x}
   \]

3. **Simplify the expression:**

   Rewrite the expression as:

   \[
   \cos x + \frac{2 \sin^2 x \cos x}{\cos^2 x - \sin^2 x}
   \]

4. **Combine terms over a common denominator:**

   The common denominator is \( \cos^2 x - \sin^2 x \). Rewrite \( \cos x \) as \( \frac{\cos x (\cos^2 x - \sin^2 x)}{\cos^2 x - \sin^2 x} \):

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

5. **Factor out \( \cos x \) from the numerator:**

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

6. **Simplify inside the parentheses:**

   \[
   \cos^2 x - \sin^2 x + 2 \sin^2 x = \cos^2 x + \sin^2 x
   \]

   And since \( \cos^2 x + \sin^2 x = 1 \), the expression simplifies to:

   \[
   \frac{\cos x \cdot 1}{\cos^2 x - \sin^2 x} = \frac{\cos x}{\cos 2x}
   \]

### Conclusion:

We have shown that:

\[
\cos x + \sin x \tan 2x = \frac{\cos x}{\cos 2x}
\]

Thus, the identity is proven.

Would you like further details on any specific steps, or do you have questions?

Here are five related questions to further explore this topic:

1. How is the double-angle formula for tangent derived?
2. Why is it helpful to use common denominators when proving trigonometric identities?
3. What are other common double-angle identities in trigonometry?
4. How can you simplify trigonometric expressions using Pythagorean identities?
5. Are there alternative methods to prove this identity?

**Tip:** Always consider expanding or simplifying trigonometric identities using known identities, such as double-angle, Pythagorean, or sum-to-product identities.

Prove the identity: cos x + sin x tan 2x = cos x / cos 2x

Explore a step-by-step solution to prove the trigonometric identity cos x + sin x tan 2x = cos x / cos 2x. This proof uses double-angle identities and Pythagorean identities, making it suitable for high school trigonometry students in Grades 10-12.

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