To solve the expression \( 7 \tan^2 x - 7 \sec^2 x + 7 \), let's proceed step by step.

### Step 1: Factor the expression
The given expression can be factored as:

\[
7(\tan^2 x - \sec^2 x) + 7
\]

### Step 2: Simplify the trigonometric identity
We know from the Pythagorean identity:

\[
\sec^2 x = 1 + \tan^2 x
\]

Substituting this into the expression:

\[
\tan^2 x - \sec^2 x = \tan^2 x - (1 + \tan^2 x)
\]

Simplifying further:

\[
\tan^2 x - \sec^2 x = \tan^2 x - \tan^2 x - 1 = -1
\]

### Step 3: Substitute back into the expression
Substituting \(-1\) into the original expression:

\[
7(-1) + 7 = -7 + 7 = 0
\]

### Final Answer
So, the value of \( 7 \tan^2 x - 7 \sec^2 x + 7 \) is \( \boxed{0} \).

Would you like more details on any step or have any further questions?

Here are some related questions to explore:
1. What is the Pythagorean identity for trigonometric functions?
2. How can you derive the identity for \(\sec^2 x\)?
3. What are the other trigonometric identities that involve \(\tan x\)?
4. How do you simplify expressions involving trigonometric functions?
5. Can you generalize this approach to other similar expressions?

**Tip:** Understanding the fundamental trigonometric identities is key to simplifying complex expressions and solving trigonometric equations.

Learn how to solve the expression 7 tan^2 x - 7 sec^2 x + 7 with a step-by-step solution. This problem involves applying trigonometric identities, including the Pythagorean identity, to simplify and evaluate the expression. Suitable for advanced high school students in Grades 11-12.

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