Find a simplified expression for cos(tan^(-1)(x/5))

Why did the scarecrow become an expert in trigonometry? Because he was outstanding in his field! Now, let's tackle your question. So we have cos(tan^(-1)(x/5)).

Let’s start by using some clown magic: tan^(-1)(x/5) can be rewritten as arctan(x/5). Now, we can use the identity cos(arctan(y)) = 1 / sqrt(1 + y^2).
Therefore, when we substitute y with x/5 in our identity, we get cos(tan^(-1)(x/5)) = 1 / sqrt(1 + (x/5)^2). Now, let's simplify the expression further:

1 / sqrt(1 + (x/5)^2)

= 1 / sqrt(1 + (x^2/25))

= 1 / sqrt((x^2 + 25) / 25)

= sqrt(25 / (x^2 + 25))

= sqrt(25) / sqrt(x^2 + 25)

= 5 / sqrt(x^2 + 25)

So there you have it! The simplified expression for cos(tan^(-1)(x/5)) is 5 / sqrt(x^2 + 25).

To find a simplified expression for cos(tan^(-1)(x/5)), we can use the trigonometric identities.

First, let's consider the triangle formed by the sides adjacent, opposite, and hypotenuse in the arctan(x/5) expression.

Let's label the sides as follows:
Opposite side: x
Adjacent side: 5

Using the Pythagorean theorem, the hypotenuse can be calculated as follows:
hypotenuse^2 = adjacent side^2 + opposite side^2
hypotenuse^2 = 5^2 + x^2
hypotenuse^2 = 25 + x^2
hypotenuse = sqrt(25 + x^2)

Now, let's use the definitions of sine, cosine, and tangent to simplify the expression further:

cos(tan^(-1)(x/5)) = cos(θ)

Since tan(θ) = opposite/adjacent = x/5, we can find the value of the adjacent side using the Pythagorean theorem:

adjacent side = sqrt(hypotenuse^2 - opposite side^2)
adjacent side = sqrt((sqrt(25 + x^2))^2 - x^2)
adjacent side = sqrt(25 + x^2 - x^2)
adjacent side = sqrt(25)
adjacent side = 5

Therefore, cos(tan^(-1)(x/5)) simplifies to cos(θ) where θ is the angle in a right triangle with an adjacent side of length 5 and a hypotenuse of length sqrt(25 + x^2).

Hence, the simplified expression for cos(tan^(-1)(x/5)) is cos(θ), where θ is the angle in a right triangle with an adjacent side of length 5 and a hypotenuse of length sqrt(25 + x^2).

To find a simplified expression for cos(tan^(-1)(x/5)), we can use the definition of the tangent inverse function (tan^(-1)(x/5)) as the angle whose tangent is equal to x/5. Let's denote this angle as θ.

We can start by drawing a right-angled triangle in the Cartesian plane, where the tangent of the angle θ is x/5. Let the perpendicular side of the triangle be x, and the base be 5.

Using the Pythagorean theorem, we can find the hypotenuse:
Hypotenuse^2 = Perpendicular^2 + Base^2
Hypotenuse^2 = x^2 + 5^2
Hypotenuse = √(x^2 + 25)

Now, let's find the cosine of the angle θ using the definition of cosine:
cos(θ) = Base / Hypotenuse
cos(θ) = 5 / √(x^2 + 25)

Therefore, the simplified expression for cos(tan^(-1)(x/5)) is 5 / √(x^2 + 25).