Let's solve each part step by step.

(a) Find the tangent line approximation to $\cos x$ at $x = \frac{\pi}{4}$:

The tangent line approximation of a function $f(x)$ at a point $x = a$ is given by: $f(x) \approx f(a) + f'(a)(x - a).$ Here, $f(x) = \cos x$ and $a = \frac{\pi}{4}$.

1. Evaluate $f(a) = \cos\left(\frac{\pi}{4}\right)$: $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.$

2. Find the derivative $f'(x) = -\sin x$ and evaluate $f'(a) = -\sin\left(\frac{\pi}{4}\right)$: $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \text{so} \quad f'(a) = -\frac{\sqrt{2}}{2}.$

3. Form the tangent line approximation: $\cos x \approx \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}).$ For rounding to three decimal places: $\cos x \approx 0.707 - 0.707(x - 0.785).$

(b) Estimate the error in the approximation for $0 \leq x \leq \frac{\pi}{2}$:

To estimate the error, we use the fact that the error for a Taylor approximation of the first degree (linear approximation) is given by: $\left| R_1(x) \right| = \frac{M}{2} |x - a|^2,$ where $M$ is an upper bound on the absolute value of the second derivative $f''(x)$ over the interval.

1. Find $f''(x) = -\cos x$: The maximum value of $|\cos x|$ on $[0, \frac{\pi}{2}]$ is 1, so $M = 1$.

2. Estimate the error for $x = \frac{\pi}{2}$: $\left| R_1\left(\frac{\pi}{2}\right) \right| \approx \frac{1}{2} \left| \frac{\pi}{2} - \frac{\pi}{4} \right|^2 = \frac{1}{2} \left(\frac{\pi}{4}\right)^2 = \frac{\pi^2}{32}.$ Calculating $\frac{\pi^2}{32} \approx 0.308$.

Thus, the error is less than 0.3 (to one decimal place).

Would you like more details, or have any questions? Here are five related questions to explore further:

1. What is the tangent line approximation for other trigonometric functions?
2. How does the error change with higher-degree Taylor approximations?
3. What is the geometric interpretation of a tangent line approximation?
4. How can we determine a tighter bound on the error?
5. What happens if we approximate $\cos x$ at points other than $\frac{\pi}{4}$?

Tip: The closer $x$ is to the point of approximation, the more accurate the tangent line approximation will be.