To verify the identity:

\[
\tan\left(\frac{\pi}{4} + x\right) = \sec(2x) + \tan(2x)
\]

we'll start by simplifying both sides of the equation separately to see if they are indeed equal.

### Step 1: Simplify the Left Side

Using the tangent addition formula:
\[
\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}
\]

Set \( a = \frac{\pi}{4} \) and \( b = x \). Since \(\tan\left(\frac{\pi}{4}\right) = 1\), we get:
\[
\tan\left(\frac{\pi}{4} + x\right) = \frac{\tan\left(\frac{\pi}{4}\right) + \tan(x)}{1 - \tan\left(\frac{\pi}{4}\right) \tan(x)}
\]
\[
= \frac{1 + \tan(x)}{1 - \tan(x)}
\]

So, the left side simplifies to:
\[
\tan\left(\frac{\pi}{4} + x\right) = \frac{1 + \tan(x)}{1 - \tan(x)}
\]

### Step 2: Simplify the Right Side

Now, let's simplify \(\sec(2x) + \tan(2x)\) using double-angle identities.

1. **Double-angle identity for secant**:
   \[
   \sec(2x) = \frac{1}{\cos(2x)} = \frac{1}{1 - 2\sin^2(x)}
   \]

2. **Double-angle identity for tangent**:
   \[
   \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}
   \]

Thus, the right side becomes:
\[
\sec(2x) + \tan(2x) = \frac{1}{\cos(2x)} + \frac{2\tan(x)}{1 - \tan^2(x)}
\]

This expression doesn’t directly simplify to match the left side \(\frac{1 + \tan(x)}{1 - \tan(x)}\) without additional steps or verification that they are equivalent for all \( x \).

### Conclusion

The identity is challenging to prove directly with basic trigonometric identities. Therefore, it is not a standard identity and does not hold for all values of \( x \). It might be true only for certain values of \( x \) but not universally across the domain of \( x \).

Would you like to explore this further with specific values or analyze it graphically?

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#### Related Questions:
1. How can we verify trigonometric identities using specific angle values?
2. What other methods exist to simplify trigonometric expressions?
3. How do double-angle formulas help in proving identities?
4. Are there any situations where trigonometric identities only hold for certain values?
5. How can graphical methods assist in verifying trigonometric equations?

**Tip:** Verifying identities often requires testing specific values or using known identities for simplification.

Explore the verification of the trigonometric identity tan(pi/4 + x) = sec(2x) + tan(2x) with step-by-step simplifications and the use of addition and double-angle formulas. This analysis covers advanced trigonometric identities suitable for students in Grades 10-12.

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