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We are given the function \( f(x) = \tan^2(x) \) and the point \( \left( \frac{\pi}{4}, 1 \right) \). The task is to find the equation of the tangent line to the graph of this function at that point.

### Steps:
1. **Find the derivative of \( f(x) = \tan^2(x) \):**
   To find the equation of the tangent line, we first need to compute the derivative of the function. We will use the chain rule:

   \[
   f(x) = \tan^2(x)
   \]
   The derivative of \( f(x) \) is:
   \[
   f'(x) = 2\tan(x) \cdot \sec^2(x)
   \]

2. **Evaluate the derivative at \( x = \frac{\pi}{4} \):**
   We substitute \( x = \frac{\pi}{4} \) into the derivative:
   \[
   f'\left( \frac{\pi}{4} \right) = 2 \tan\left( \frac{\pi}{4} \right) \cdot \sec^2\left( \frac{\pi}{4} \right)
   \]
   Since \( \tan\left( \frac{\pi}{4} \right) = 1 \) and \( \sec\left( \frac{\pi}{4} \right) = \sqrt{2} \), we get:
   \[
   f'\left( \frac{\pi}{4} \right) = 2 \times 1 \times 2 = 4
   \]

3. **Write the equation of the tangent line:**
   The equation of a tangent line is given by the point-slope form:
   \[
   y - y_1 = m(x - x_1)
   \]
   Here, \( m = 4 \) (the slope), \( x_1 = \frac{\pi}{4} \), and \( y_1 = 1 \). Substituting these values into the equation:
   \[
   y - 1 = 4 \left( x - \frac{\pi}{4} \right)
   \]

4. **Simplify the equation:**
   \[
   y - 1 = 4x - \pi
   \]
   \[
   y = 4x - \pi + 1
   \]
   Thus, the equation of the tangent line is:
   \[
   y = 4x - \pi + 1
   \]

Would you like further details or clarifications on any step?

### Related questions:
1. How is the chain rule applied to functions like \( \tan^2(x) \)?
2. What is the geometric meaning of the tangent line to a curve?
3. How does evaluating a derivative at a point relate to the slope of the tangent line?
4. Can we always use point-slope form to find the equation of a tangent line?
5. What happens if we change the function to something like \( \cot^2(x) \)?

**Tip:** The chain rule is essential when differentiating compositions of functions like \( \tan^2(x) \), where the outer function is a square and the inner function is \( \tan(x) \).

Find an equation of the tangent line to the graph of the function f(x) = tan^2(x) at the point (π/4, 1).

Learn how to find the tangent line to the function f(x) = tan^2(x) at the point (π/4, 1) using derivatives and the chain rule. This solution includes step-by-step calculations, applying key calculus concepts like the chain rule and point-slope form for tangent lines.

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