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To prove the identity \((\tan^2 x + 1)\cos 2x = 2 - \sec^2 x\), we can follow these steps:

### Step 1: Simplify \((\tan^2 x + 1)\)
Using the Pythagorean identity, we have:
\[
\tan^2 x + 1 = \sec^2 x
\]

### Step 2: Substitute into the left side
Substituting \(\sec^2 x\) into the equation:
\[
(\sec^2 x)\cos 2x
\]

### Step 3: Use the double angle identity for cosine
The identity for \(\cos 2x\) is:
\[
\cos 2x = 2\cos^2 x - 1
\]

### Step 4: Substitute \(\cos 2x\) into the expression
\[
(\sec^2 x)(2\cos^2 x - 1)
\]

### Step 5: Simplify the expression
Since \(\sec^2 x = \frac{1}{\cos^2 x}\), we substitute:
\[
\left(\frac{1}{\cos^2 x}\right)(2\cos^2 x - 1) = 2 - \frac{1}{\cos^2 x}
\]

### Step 6: Use the identity \(\sec^2 x = \frac{1}{\cos^2 x}\)
Therefore, the expression becomes:
\[
2 - \sec^2 x
\]

This completes the proof.

Would you like more details or have any questions?

1. How do I use trigonometric identities in proofs?
2. What are some common trigonometric identities?
3. How do double angle identities work?
4. Can you explain the Pythagorean identities?
5. What are some tips for simplifying trigonometric expressions?

**Tip:** Always start with known identities and simplify step-by-step.

Learn how to prove the trigonometric identity (tan^2 x + 1)cos 2x = 2 - sec^2 x step-by-step. This problem involves using trigonometric identities such as the Pythagorean identity and the double angle identity. Suitable for advanced high school students and those studying trigonometry.

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