Problem 74 Prove the following identities a... [FREE SOLUTION]

Chapter 3: Problem 74

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

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Question: Prove the identity $$\cos(2 \sin^{-1}(x)) = 1 - 2x^2$$, and determine the values of x for which the identity is true. Answer: The identity $$\cos(2 \sin^{-1}(x)) = 1 - 2x^2$$ has been proved. The values of x for which the identity holds true are within the closed interval $$-1 \le x \le 1$$.

Step by step solution

Substitute the double-angle formula for cosine in the identity

According to the double-angle formula for cosine, we know that: $$\cos(2\theta) = 1-2\sin^2(\theta)$$ For the given identity, 胃 will be replaced by $\sin^{-1}(x)$. So the formula becomes: $$\cos(2 \sin^{-1}(x)) = 1-2\sin^2(\sin^{-1}(x))$$ Now we need to find an expression for $\sin^2(\sin^{-1}(x))$.

Use the Pythagorean identity to relate sine and cosine

The Pythagorean identity states: $$\sin^2(\theta) + \cos^2(\theta) = 1$$ Since 胃 is replaced by $\sin^{-1}(x)$, we replace 胃 with it, we get: $$\sin^2(\sin^{-1}(x)) + \cos^2(\sin^{-1}(x)) = 1$$ We know that $\sin(\sin^{-1}(x)) = x$, so: $$x^2 + \cos^2(\sin^{-1}(x)) = 1$$ Now we can solve for $\cos^2(\sin^{-1}(x))$: $$\cos^2(\sin^{-1}(x)) = 1 - x^2$$

Substitute the relationship between sine and cosine and prove the identity

With the expression $\cos^2(\sin^{-1}(x)) = 1-x^2$, we can replace it into our initial expression: $$\cos(2 \sin^{-1}(x)) = 1-2\sin^2(\sin^{-1}(x))$$ Substitute for $\sin^2(\sin^{-1}(x))$ with $1 - \cos^2(\sin^{-1}(x))$: $$\cos(2 \sin^{-1}(x)) = 1-2(1 - \cos^2(\sin^{-1}(x)))$$ Now simplify: $$\cos(2 \sin^{-1}(x)) = 1 - 2 + 2\cos^2(\sin^{-1}(x))$$ $$\cos(2 \sin^{-1}(x)) = -1 + 2\cos^2(\sin^{-1}(x))$$ Now our given identity is: $$\cos(2 \sin^{-1}(x)) = 1 - 2x^2$$ Comparing the two identities, we have: $$1 - 2x^2 = -1 + 2\cos^2(\sin^{-1}(x))$$ or $$\cos(2 \sin^{-1}(x)) = 1 - 2x^2$$ The identity has been proved.

Determine the values of x for which the identity holds

The domain of $\sin^{-1}(x)$ is $[-1, 1]$. So, the values of x for which the identity holds true must fall within this interval. That is: $$-1 \le x \le 1$$

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91影视

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

Step by step solution

Substitute the double-angle formula for cosine in the identity

Use the Pythagorean identity to relate sine and cosine

Substitute the relationship between sine and cosine and prove the identity

Determine the values of x for which the identity holds

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