What's the value of cos(arctan(x))?

Question

Evaluate cos(arctan(x)).

Answer 1

cos(arctanx) = 1/(sqr(x^2+1)

construct a right triangle

with height=x, base = 1, tanH = x/1 = x

arctanx = H= angle opposite side h=height

cosH = base/hypotenuse = 1/side opposite 90 degrees

= 1/sqr(x^2+1)

Answer 2

Draw a sketch of a triangle. First consider the inside function: arctan(x)

Remember that the arctan function (or any other "arc" expressions) will produce an angle. Let's call the angle θ. And since the tangent function is a ratio of opposite to adjacent, let's say that we are looking for arctan(x/1) or, in other words, its the angle that has a tangent of x/1 therefore, in our triangle sketch, the opposite side is "x" and the adjacent side is "1":

Now, find the cosine of the angle θ in your sketch. To do that, since cos(θ) = adjacent/hypotenuse, you must first find the hypotenuse. Use the Pythagorean Theorem:

h² = 1² + x² (where h = the hypotenuse)

h = √(1 + x²)

So the cosine is 1/√(1 + x²)

You may wish to rationalize this. If so, multiply by √(1 + x²)/√(1 + x²) to get √(1 + x²)/(1 + x²)

Answer 3

We need to evaluate cos(arctan(x)).

We perform a substitution:

y = arctan(x)

tan(y) = x

Solution 1

We now need to find cos(y).

Now,

tan(y) = sin(y)/cos(y) = x/1

We imagine a triangle with angle y in one corner and angle 90. The side adjacent to angle y (cos y) is set to length 1, and the opposite (sin y) is set to x.

The hypotenuse of this triangle is:

x^2+1 = h^2

sqrt(x^2+1) = h

(sqrt is the square root function.)

cos(y) is the adjacent side of length 1 over the hypotenuse that we've just calculated.

cos(y) = 1/sqrt(x^2+1)

So:

cos(arctan(x)) = 1/sqrt(x^2+1)

Solution 2

We now need to find cos(y).

Now,

tan(y) = sin(y)/cos(y) = x

Can we eliminate sin(y) and just have cos(y)?

sin^2(y)/cos^2(y) = x^2

If we can add cos^2(y) to sin^2(y), it can be simplified.

sin^2(y)/cos^2(y) + cos^2(y)/cos^2(y) = x^2 + cos^2(y)/cos^2(y)

[sin^2(y) + cos^2(y)]/cos^2(y) = x^2 + 1

1/cos^2(y) = x^2+1 (because of the general identity that sin^2(a)+cos^2(a)=1).

1/(x^2+1) = cos^2(y)

1/sqrt(x^2+1) = cos(y)

cos(y) = 1/sqrt(x^2+1)

So:

cos(arctan(x)) = 1/sqrt(x^2+1)