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Chapter 5: Problem 83

Verify the identity. $$\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}$$

Short Answer

Expert verified
The identity \(\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}\) is verified as true.

Step by step solution

01

Understand the inverse sine function

Firstly, let's denote the inverse sine function of \(x\) as \(\theta\). So, we have \(\sin ^{-1} x = \theta\). This means that \(\sin(\theta) = x\).
02

Use Pythagorean identity

Recall the Pythagorean identity for sine and cosine, \(\sin^2(\theta) + \cos^2(\theta) = 1\). Since we know that \(\sin(\theta) = x\), we can substitute \(x\) into this equation: \(x^2 + \cos^2(\theta) = 1\).
03

Solve for \(\cos(\theta)\)

Now, let's solve for \(\cos(\theta)\) by isolating it on one side of the equation. Subtract \(x^2\) from both sides to get \(\cos^2(\theta) = 1 - x^2\). The positive square root of this is \(\sqrt{1 - x^2}\), which is equal to \(\cos(\theta)\). So, \(\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}\).

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