Question

Prove the following identities and give the values of \(x\) for which they are true. $$\cos \left(2 \sin ^{-1} x\right)=1-2 x^{2}$$

Short answer

Question: Prove the trigonometric identity \(\cos(2\sin^{-1}(x))=1-2x^2\) and determine the values of \(x\) for which the identity holds true. Answer: The identity \(\cos(2\sin^{-1}(x))=1-2x^2\) holds true for all values of \(x\) in the interval \([-1, 1]\).

Step-by-step solution

Rewrite using a variable substitution.

Let \(y=\sin^{-1}(x)\). Then \(\sin(y) = x\). Our goal is to find \(\cos (2y)\).

Use trigonometric identity.

Recall the double-angle formula for cosine: \(\cos (2y) = 1- 2\sin^2 y\). We need to rewrite this in terms of \(x\). Since we know \(\sin(y) = x\), we can write it as: $$\cos(2y) = 1- 2x^2$$

Substitute back to the original variable.

Now, we substitute the original variable back in. Recall that \(y = \sin^{-1}(x)\), so we write the expression as: $$\cos(2\sin^{-1}(x)) = 1 - 2x^2$$ Hence, we have proven the identity.

Determine the valid range of x.

The trigonometric function \(\sin^{-1}(x)\) is only defined for \(x\in[-1, 1]\). Therefore, the identity is true for all \(x\) in the interval \([-1, 1]\): $$x\in[-1,1]$$

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when their trigonometric values are known. They play a significant role in trigonometry because they allow us to work backward from a ratio to an angle. The notation \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), and \(\tan^{-1}(x)\) are commonly used, where 'sin-inverse', 'cos-inverse', and 'tan-inverse' operate in a way that maps the trigonometric ratio back to an angle. For example, if you know that \(\sin(\theta) = x\), then \(\theta = \sin^{-1}(x)\). In the context of the problem, we set \(y = \sin^{-1}(x)\), representing an angle \(y\) whose sine value is \(x\). Thus, \(\sin(y) = x\), which is a key substitution that simplifies the expression from the given identity and makes it easier to work with.

Double-Angle Formulas

The double-angle formulas are crucial identities in trigonometry that express trigonometric functions of double angles (like \(2\theta\)) in terms of single angles. These formulas help simplify expressions and solve trigonometric equations. For cosine, the double-angle formula is \(\cos(2\theta) = 1 - 2\sin^2(\theta)\), which can also be written as \(\cos^2(\theta) - \sin^2(\theta)\) or \(2\cos^2(\theta) - 1\).In our problem, once we identified \(y = \sin^{-1}(x)\), we used the double-angle formula \(\cos(2y) = 1 - 2\sin^2(y)\) and substituted \(\sin(y) = x\). This made the identity become \(\cos(2\sin^{-1} x) = 1 - 2x^2\), using the double-angle formula in a smart way to transform the equation.

Valid Range of Trigonometric Functions

The valid range of a trigonometric function refers to the set of values for which the function is defined. For inverse trigonometric functions, understanding their range is important to ensure the solutions are correct and valid.- For \(\sin^{-1}(x)\), the values of \(x\) must lie within \([-1, 1]\), corresponding to angles from \(-\pi/2\) to \(\pi/2\).- For \(\cos^{-1}(x)\), \(x\) also needs to be within \([-1, 1]\), with results between \(0\) and \(\pi\).- For \(\tan^{-1}(x)\), \(x\) can be any real number, and the range is \((-\pi/2, \pi/2)\).In the exercise, the range \([-1, 1]\) for \(\sin^{-1}(x)\) ensures that the identity holds for all admissible values of \(x\). Hence, determining this valid range is crucial for correctly applying the identity across all necessary scenarios.