What are the steps involved in reducing cos(arctan(2/5))? I was completely stumped on this problem during a test. I have googled the answer and came across this link: http://answers.yahoo.com/question/index?qid=20110310212347AAJ21sr
However, I still do not understand.
Also, I used wolframalpha to discover the answer is: 5/sqrt(29) radians
http://www.wolframalpha.com/input/?i=cos%28arctan%282%2F5%29%29
Any help understanding how to solve this type of problem will be greatly appreciated
 A: Let's try walking through this piece by piece. 
First, let's try to understand what these functions mean and do. In general, trigonometric functions take in an ${\bf angle}$ and return a ${\bf ratio}$. In the case of, say, $\cos$ for example, the functions $\cos(x)$ takes in an angle $x$ and returns the ratio of the adjacent side over the hypotenuse of the triangle inscribed on the unit circle with angle $x$. 
Inverse trigonometric functions do just the opposite. They take as an input a given ratio and return an associated angle. Let's see if we can use this to solve the problem.
The first step involves using $\arctan(\frac{2}{5}))$. This returns an angle, but we'll see that we don't need to know the value of this angle - we just need to use the information conferred by the ratio to construct the appropriate triangle. What triangle are we talking about here? Well, the input ratio is $\frac{2}{5}$ and we're looking at $\arctan$. Since $\tan$ gives the angle associated with the ratio of the opposite side over the adjacent side, we want to construct a triangle with opposite side of length $2$ and adjacent side of length $5$. Since we may need to know the length of the hypotenuse of this triangle, let's calculate it. The pythagorean theorem says it will be of length $\sqrt{2^{2} + 5^{2}} = \sqrt{29}$. 
Good! Now, let's solve the problem. We want to take find $\cos(\arctan(\frac{2}{5}))$, so let's put that into words now. We want to find the ratio of the length of the adjacent side over the length of the hypotenuse of the triangle whose opposite side and adjacent side have a length ratio of $\frac{2}{5}$. That's a mouthful, but now we know what we want to find! Simply take the length of the adjacent side and divide it by the length of the hypotenuse. In this case, our answer is $\frac{5}{\sqrt{29}}$. 
A: $\arctan\dfrac25$ is the angle whose tangent is $\dfrac25$.  Call that angle $\theta$.  So $\tan\theta=\dfrac25$.  Since $\tan=\dfrac{\text{opposite}}{\text{adjacent}}$, we could have $\text{opposite}=2$ and $\text{adjacent}=5$ (or we could have $\text{opposite}=20$ and $\text{adjacent}=50$, etc.).
If $\text{opposite}=2$ and $\text{adjacent}=5$, then the Pythagorean theorem gives you the hypotenuse, and then you can say $\cos=\dfrac{\text{adjacent}}{\text{hypotenuse}}$.
