/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 Verify the identity. $$\cos \l... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 48

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 after substituting \(\sin 胃 = x\) into the Pythagorean identity and simplifying.

Step by step solution

01

Assign the value of the inverse sine function

Let \(\sin ^{-1} x = 胃\). This implies that \(x=\sin 胃\) following the definition of the inverse sine function.
02

Use the Pythagorean identity

Knowing that \(x^2 = \sin^2 胃\), we can use the Pythagorean identity \(1 = \sin^2 胃 + \cos^2 胃\) to express cosine in terms of sine. By rearranging, we get \(\cos^2 胃 = 1 - \sin^2 胃\) which implies \(\cos 胃 = \sqrt{1 - \sin^2 胃}\).
03

Substitute back into the original equation

Substitute \(\sin 胃 = x\) into \(\cos 胃 = \sqrt{1 - \sin^2 胃}\), giving \(\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}\) as required.

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