Simplify the expression cot (cos−1(x)).
Not sure how to do this.
cos−1(x) refers to an angle, let's call it Ø , so that
cosØ = x = x/1
so sketch a right-angled triangle, label the base angle Ø, it's adjacent side as 1, and its hypotenuse as x
then the other side is √(1 - x^2)
and cotØ = x/√(1 - x^2)
thus cot (cos−1(x)) = x/√(1 - x^2)
To simplify the expression cot(cos⁻¹(x)), we can use the definition of the inverse cosine function.
The inverse cosine function, cos⁻¹(x), returns the angle whose cosine is x.
So, if we take the inverse cosine of x, we get an angle θ such that cos(θ) = x.
Now, cotangent is the ratio between the adjacent side (next to the angle) and the opposite side (opposite to the angle) in a right-angled triangle.
To find the cotangent of cos⁻¹(x), we need to consider a right triangle with an angle θ such that cos(θ) = x.
Let's assign the opposite side as x (since sin(θ) = opposite/hypotenuse and hypotenuse is typically 1), the adjacent side as 1, and the hypotenuse as √(x² + 1) (using the Pythagorean theorem).
Now, the cotangent of θ is defined as the ratio of the adjacent side to the opposite side:
cot(θ) = adjacent/opposite
= 1/x
Thus, we can simplify the expression cot(cos⁻¹(x)) to 1/x.
To simplify the expression cot (cos−1(x)), we need to make use of trigonometric identities. Let's break down the steps to simplify the given expression.
Step 1: Recall the definition of the inverse cosine function (cos−1(x)):
The inverse cosine function, cos−1(x) (also denoted as arccos(x)), returns the angle θ (in radians) whose cosine is x. In other words, cos(θ) = x.
Step 2: Identify the relationship between cosine (cos) and cotangent (cot):
By using the trigonometric identity, we can relate cotangent (cot) and cosine (cos):
cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ)
Step 3: Substitute the value from Step 1 into the expression cot (cos−1(x)):
We substitute θ = cos−1(x) in the expression cot(θ), which gives us:
cot (cos−1(x)) = cot (cos−1(x)) = cos (cos−1(x)) / sin (cos−1(x))
Step 4: Evaluate cos (cos−1(x)) and sin (cos−1(x)):
Since θ = cos−1(x), cos(θ) = x and sin(θ) can be determined using the Pythagorean identity:
sin(θ) = √(1 - cos^2(θ)) = √(1 - x^2)
By substituting these values into the expression, we get:
cot (cos−1(x)) = cos (cos−1(x)) / sin (cos−1(x)) = x / √(1 - x^2)
Therefore, the simplified expression for cot (cos−1(x)) is x / √(1 - x^2).