To simplify \(\cos(\arctan(x))\), we can follow these steps:

1. Let \(\theta = \arctan(x)\). This means that \(\tan(\theta) = x\), or equivalently, \(\frac{\sin(\theta)}{\cos(\theta)} = x\).
   
2. From \(\tan(\theta) = x\), we can express \(\sin(\theta)\) and \(\cos(\theta)\) using a right triangle:
   - Imagine a right triangle where the opposite side (relative to angle \(\theta\)) is \(x\) and the adjacent side is 1, since \(\tan(\theta) = \frac{x}{1}\).
   - Using the Pythagorean theorem, the hypotenuse of this triangle is \(\sqrt{x^2 + 1}\).

3. Now, using the definition of \(\cos(\theta)\) in a right triangle:
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{x^2 + 1}}.
   \]

Thus, \(\cos(\arctan(x)) = \frac{1}{\sqrt{x^2 + 1}}\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you simplify \(\sin(\arctan(x))\)?
2. What is the general process for simplifying \(\cos(\arcsin(x))\)?
3. How does the Pythagorean theorem help in solving trigonometric expressions involving inverse functions?
4. Can you find \(\cos(\arctan(x) + \frac{\pi}{4})\)?
5. How would you differentiate \(\cos(\arctan(x))\)?

**Tip:** Inverse trigonometric functions can often be simplified by visualizing them in terms of right triangles.

Discover how to simplify cos(arctan(x)) using trigonometric identities and the Pythagorean theorem. This detailed guide walks through key concepts such as right triangles, trigonometric functions, and inverse functions, making it ideal for high school students in Grades 10-12.

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