Draw a right triangle to simplify the given expressions....

Draw a right triangle to simplify the given expressions. Assume $x>0$
$$\cos \left(2 \sin ^{-1} x\right)\left(\text {Hint}: \text { Use } \cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta .\right)$$

Key Concepts

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is fundamental in trigonometry. It allows one to derive one trigonometric ratio from another and is used in simplifying expressions, particularly when a triangle is constructed and one side needs to be determined using the relationships between the sides.

Double Angle Formulas

Double angle formulas, like cos 2? = cos²? ? sin²?, are identities that express trigonometric functions of doubled angles in terms of functions of a single angle. These formulas are crucial for reducing complex trigonometric expressions and solving equations by relating angles in multiple ways.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the inverse sine function, allow us to determine an angle when the value of a trigonometric ratio is known. They are essential for converting between angle measures and ratios, which permits the analysis of trigonometric equations and expressions.

Right Triangle Representation

Drawing a right triangle is a visual method for linking trigonometric ratios with side lengths. By constructing a triangle based on the given trigonometric relationships, one can express sine, cosine, and other functions in terms of the sides of the triangle, simplifying the evaluation of expressions.

Transcript

00:01 To start off this problem, i'm going to define the arc sign of x to be some angle theta, since the arc sign takes in a length, spits out an angle.

00:16 I'm going to use this theta by taking the sign on both sides to say that the sign of some angle theta is equal to x, where theta, in this case, has to range between negative pi over two and pi over two.

00:33 From this, we can use the definition of the sign of theta, which is that it is the y side of a right triangle divided by the hypotenuse to draw a right triangle representing the situation.

00:49 This is our angle theta, this is our y, this is our r, and this is our x.

00:58 And one thing you can notice is that x here, the sign of theta being able to x, is just x divided by 1.

01:08 So therefore we can set r as the denominator being able to 1 and y being equal to x.

01:16 And i'm going to instead of call the x part x, i'm going to call it x dash to not confuse ourselves.

01:25 Therefore, if the arc sign of x is theta, this becomes the cosine of 2 theta.

01:33 And here we can use the identity that the cosine of 2 theta is equal to the cosine squared of theta plus the sine squared of theta.

01:46 I'm actually going to rewrite this to make it a little simpler for us in the long run by saying that this is the cosine of theta all squared plus the sign of theta all squared, which is the same thing as what we had before.

02:01 So now this represents the cosine of theta and sitem theta of this triangle right here...

Please give Ace some feedback