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Chapter 5: Problem 47

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

Short Answer

Expert verified
The given identity \( \tan(\sin^{-1} x) = \frac{x}{\sqrt{1-x^{2}}} \) is verified as true after simplification.

Step by step solution

01

Simplifying the Left Hand Side

Let the angle \( \sin^{-1}x \) be denoted as \( y \). This means that \( x = \sin y \). So the left hand side can be rewritten as \( \tan y \).
02

Expressing \( \tan y \) in Terms of \( x \)

The tan function can be expressed as the ratio of sine to cosine, namely \( \tan y = \frac{\sin y}{\cos y} \). If \( x = \sin y \), then by applying the Pythagorean identity \( \cos y = \sqrt{1 - \sin^2 y} \), we have \( \cos y = \sqrt{1 - x^2} \), resulting in \( \tan y = \frac{x}{\sqrt{1 - x^2}} \).

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