cosarctanx=

The expression (cos(arctan(x))) is a bit tricky because the cosine of the arctangent of (x) is not simply (x). To understand this, let's consider the following:

The arctangent function, (arctan(x)), gives us the angle whose tangent is (x). In other words, if (theta = arctan(x)), then (tan(theta) = x).

Now, if we take the cosine of this angle, (cos(theta)), we need to consider the unit circle. On the unit circle, the cosine of an angle is the (x)-coordinate of the point where the terminal side of the angle intersects the circle.

When we have (tan(theta) = x), the angle (theta) lies in either the first or third quadrant if (x) is positive, or in the second or fourth quadrant if (x) is negative. The cosine of an angle in the first or third quadrant is positive, and in the second or fourth quadrant, it is negative.

So, the expression (cos(arctan(x))) is actually the (x)-coordinate of the point on the unit circle where the terminal side of the angle (theta) intersects the circle, which is the same as the value of (x) itself. However, we must consider the quadrant in which the angle lies.

If (x > 0), then (cos(arctan(x)) = x).

If (x < 0), then (cos(arctan(x)) = -x).

Therefore, the expression (cos(arctan(x))) simplifies to (x) if (x) is non-negative, and (-x) if (x) is negative. In mathematical notation, this can be written as:

[

cos(arctan(x)) =

begin{cases