https://file.aiquickdraw.com/mathbot/file/1725360306760n43hcsa1.jpg

To solve this problem, we start by using the given information:

\[
\tan \alpha = 3
\]

From this, we can find the values of \(\sin \alpha\) and \(\cos \alpha\) using a right triangle approach or trigonometric identities.

### Step 1: Find \(\sin \alpha\) and \(\cos \alpha\)

Since \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = 3\), we can use the identity:

\[
\tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = 3 = \frac{3}{1}
\]

So, consider a right triangle where the opposite side to angle \(\alpha\) is 3, and the adjacent side is 1. The hypotenuse \(h\) is:

\[
h = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]

Thus, 

\[
\sin \alpha = \frac{3}{\sqrt{10}}, \quad \cos \alpha = \frac{1}{\sqrt{10}}
\]

### Step 2: Find \(\cos 4\alpha\)

Using the double-angle identity:

\[
\cos 2\alpha = 2\cos^2 \alpha - 1
\]

First, calculate \(\cos 2\alpha\):

\[
\cos^2 \alpha = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10}
\]

\[
\cos 2\alpha = 2 \times \frac{1}{10} - 1 = \frac{2}{10} - 1 = \frac{2}{10} - \frac{10}{10} = -\frac{4}{5}
\]

Now, use the double-angle identity again to find \(\cos 4\alpha\):

\[
\cos 4\alpha = 2\cos^2 2\alpha - 1
\]

\[
\cos^2 2\alpha = \left(-\frac{4}{5}\right)^2 = \frac{16}{25}
\]

\[
\cos 4\alpha = 2 \times \frac{16}{25} - 1 = \frac{32}{25} - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25}
\]

### Conclusion

The correct answer is:

\[
\boxed{\frac{7}{25}}
\]

Would you like more details or have any questions?

### Related Questions:

1. How do we find \(\sin 4\alpha\) using the same approach?
2. What is the value of \(\tan 2\alpha\) based on \(\tan \alpha = 3\)?
3. How can we use the Pythagorean identity to verify the values of \(\sin \alpha\) and \(\cos \alpha\)?
4. What would the answer be if \(\tan \alpha = \frac{1}{3}\)?
5. Can we solve for \(\sin 2\alpha\) directly from \(\tan \alpha\)?

### Tip:

When dealing with multiple angle identities, it's often helpful to use double-angle formulas repeatedly, as shown, to break down the problem step by step.

Learn how to solve a trigonometry problem involving finding the cosine of quadruple angle using double-angle identities and right triangle trigonometry. Suitable for high school students studying trigonometry.

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