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To find the equation of the tangent line to the curve \( y = 6 \sin x + 7 \sec x \) at \( x = \pi \), we will proceed as follows:

### 1. **Find the derivative of the function:**

The derivative \( \frac{dy}{dx} \) will give us the slope of the tangent line.

- Derivative of \( 6 \sin x \) is \( 6 \cos x \).
- Derivative of \( 7 \sec x \) is \( 7 \sec x \tan x \).

Thus, the derivative is:

\[
\frac{dy}{dx} = 6 \cos x + 7 \sec x \tan x
\]

### 2. **Evaluate the derivative at \( x = \pi \):**

Now, substitute \( x = \pi \) into the derivative to find the slope at that point.

- \( \cos(\pi) = -1 \)
- \( \sec(\pi) = -1 \) (since \( \sec(x) = \frac{1}{\cos(x)} \))
- \( \tan(\pi) = 0 \)

Therefore:

\[
\frac{dy}{dx}\bigg|_{x = \pi} = 6(-1) + 7(-1)(0) = -6
\]

So, the slope of the tangent line at \( x = \pi \) is \( m = -6 \).

### 3. **Find the point on the curve at \( x = \pi \):**

To find the equation of the tangent line, we need the point on the curve at \( x = \pi \). Substitute \( x = \pi \) into the original function:

\[
y = 6 \sin(\pi) + 7 \sec(\pi)
\]

Since \( \sin(\pi) = 0 \) and \( \sec(\pi) = -1 \), we get:

\[
y = 6(0) + 7(-1) = -7
\]

Thus, the point is \( (\pi, -7) \).

### 4. **Equation of the tangent line:**

The equation of the tangent line is given by the point-slope form:

\[
y - y_1 = m(x - x_1)
\]

Substitute \( m = -6 \), \( x_1 = \pi \), and \( y_1 = -7 \):

\[
y + 7 = -6(x - \pi)
\]

Simplify:

\[
y = -6x + 6\pi - 7
\]

Thus, the equation of the tangent line is:

\[
y = -6x + 6\pi - 7
\]

Would you like further details on this solution, or do you have any questions? Here are five related questions you might find interesting:

1. How do you find the derivative of trigonometric functions like \( \sec(x) \)?
2. What is the general form of a tangent line equation?
3. How does the derivative represent the slope of a function at a given point?
4. Can you apply this method to other points on the curve?
5. How do you handle more complex trigonometric derivatives?

**Tip**: Always evaluate trigonometric functions like sine, cosine, and secant carefully at special angles like \( \pi \), \( \frac{\pi}{2} \), etc., as their values can simplify the calculations significantly.

Find the equation of the tangent line to y = 6 sin x + 7 sec x at x = π.

This solution demonstrates how to find the equation of the tangent line to the curve y = 6 sin x + 7 sec x at x = π. It covers key concepts like derivatives of trigonometric functions and applying the point-slope form for the tangent line.

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