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Chapter 3: Problem 73

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

Short Answer

Expert verified
Question: Prove the identity involving the inverse sine function and cosine function: $$\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}$$ and determine the range of values for x. Answer: The identity $$\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}$$ holds true for all values of x within the closed interval of \([-1, 1]\).

Step by step solution

01

Observe the problem and rearrange the expression

Given the expression: $$\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}$$ Our goal is to manipulate the expression in such a way that it resembles the Pythagorean identity, which is: $$\sin^2(\theta)+\cos^2(\theta)=1$$ To begin this process, let's first introduce a new variable, such as \(\theta\) and set: $$\sin^{-1}(x) = \theta$$ Now, this means that: $$\sin(\theta) = x$$ Our goal is to find the value of \(\cos(\theta)\) and show that it is equal to \(\sqrt{1-x^2}\).
02

Use Pythagorean identity to relate sine and cosine

Now that we have replaced the inverse sine function, we can find the value of \(\cos(\theta)\). We know that the Pythagorean identity states: $$\sin^2(\theta) + \cos^2(\theta) = 1$$ Let's find the value of \(\cos^2(\theta)\) by solving this equation: $$\cos^2(\theta) = 1 - \sin^2(\theta)$$ We know that \(\sin(\theta) = x\), so let's plug it in: $$\cos^2(\theta) = 1 - x^2$$
03

Determine the value of \(\cos(\theta)\)

Now, let's find the value of \(\cos(\theta)\): $$\cos(\theta) = \sqrt{1-x^2}$$ We can now state that: $$\cos\left(\sin^{-1}(x)\right) = \cos(\theta) = \sqrt{1-x^2}$$
04

Determine the range of values for x

After proving the identity, we need to determine the values of x for which the identity is true. Since we are dealing with the inverse sine function, the range of values for x must be within the domain of \(\sin^{-1}(x)\). The standard domain for \(\sin^{-1}(x)\) is: $$-1 \le x \le 1$$ Therefore, the identity holds true for all values of x within the closed interval of \([-1, 1]\).

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