How would you write cos(tan^-1u) as an algebraic expression in u?

Question

How would you write cos(tan^-1u) as an algebraic expression in u?

I know the answer, and I've seen tutorial on how to do it using pathagorus, but how would you do this using identities?

Answer 1

draw the triangle

In the triangle, label the sides 1 and u, and the hypotenuse √(1+u^2)

the angle tan^-1(u) is opposite the "u" side, since opposite/adjacent = u/1 = u

So, the cosine of that angle is adjacent/hypotenuse = 1/√(1+u^2)

You can also note that

cosθ = 1/secθ
sec^2θ = 1+tan^2θ, so

cosθ = 1/√(1+tan^2 θ)

But θ = tan^-1(u), so tanθ = u and we have

cosθ = 1/√(1+u^2)

I have always easier to carefully draw the triangle and label the sides. Then all the trig functions just drop right out.

Answer 2

To express cos(tan^(-1)(u)) in terms of u using trigonometric identities, we can start by considering a right triangle. Let's assume a right triangle with an angle A such that tan(A) = u.

Using the properties of trigonometry, we can write:
tan(A) = u
And, we know that tan(A) = opposite/adjacent sides of the triangle.

Let's label the opposite side as x (in line with angle A) and the adjacent side as 1 (since the hypotenuse for a right triangle is conventionally taken as 1).

So, from the given information, we have:
tan(A) = u = x/1 = x

Now, we can use the Pythagorean identity to find the value of the hypotenuse. The Pythagorean identity states:
sin^2(A) + cos^2(A) = 1

Using similar triangles and substituting the values, we have:
(1^2 + x^2)/1 = 1
1 + x^2 = 1
x^2 = 0

Since x^2 = 0, we find that x = 0.

Now, knowing the value of cos(A) = adjacent/hypotenuse, we can substitute the values into the expression we need to evaluate:
cos(tan^(-1)(u)) = cos(A) = 1/1 = 1

So, the algebraic expression of cos(tan^(-1)(u)) in terms of u is 1.

Answer 3

To write cos(tan^-1(u)) as an algebraic expression in u using trigonometric identities, we can make use of the following identity:

tan^-1(u) = arctan(u)

We can then substitute arctan(u) into the original expression:

cos(arctan(u))

Now, let's proceed using the following identity:

cos(arctan(u)) = 1 / √(1 + u^2)

So, we can rewrite the expression as:

1 / √(1 + u^2)

Therefore, cos(tan^-1(u)) can be written algebraically as 1 / √(1 + u^2).