Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The trigonometric identity \(\cos(\sin^{-1} x)=\sqrt{1-x^{2}}\) is successfully verified.

Step 1: Understand the inverse sine function

The inverse sine function, \(\sin^{-1}(x)\), gives us an angle. According to the definition, if \(\sin^{-1}(x)=y\), we mean to say that the sine of angle \(y\) equals \(x\). Meaning, \(y = \sin^{-1}(x)\) is equivalent to \(x = \sin(y)\). This might help to visualize the problem and convert it into something familiar.

Step 2: Apply the Pythagorean identity

The Pythagorean identity in trigonometry states that \(\sin^2 y + \cos^2 y = 1\). If \(x = \sin(y)\), square both sides we have \(x^{2} = \sin^{2}(y)\). Then rearrange the Pythagorean identity to demonstrate \(\cos(y)\) in terms of \(x\), we get \(\cos^2 y = 1 - \sin^2 y\) which equals to \(1 - x^{2}\).

Step 3: Getting the final answer

Take the square root of both sides of the equation \(\cos^2 y = 1 - x^{2}\) which gives us \(\cos y = \sqrt{1 - x^{2}}\) or \(-\cos y = \sqrt{1 - x^{2}}\). But in the quadrant where \(\sin^{-1}(x)\) is defined, the cosine function is positive, so we can ignore the negative root and our answer for \(\cos y\) is \(\sqrt{1 - x^{2}}\). So, \(\cos(\sin^{-1} x) = \sqrt{1 - x^{2}}\) has been verified.