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Chapter 5: Problem 126

If \(y=\sin ^{-1} x\), prove that \(\left(1-x^{2}\right) y_{2}-x y_{1}=0\)

Short Answer

Expert verified
The original trigonometric identity is proven valid after differentiating the function, \(y=\sin ^{-1} x\), twice, and substituting these values into the equation. The resulting expression simplifies to zero, validating the original identity.

Step by step solution

01

Find the First Derivative

To find the first derivative of the function \(y=\sin ^{-1} x\), recall the derivative of the inverse sine function, which is \(y_{1} = \frac{1}{\sqrt{1-x^{2}}}\). Thus, the first derivative of \(y\) is \(y_{1} = \frac{1}{\sqrt{1-x^{2}}}\).
02

Find the Second Derivative

Now, compute the second derivative \(y_{2}\) by differentiating \(y_{1}\) again. Upon the derivative rule for a function that expresses \(x\) in the denominator, we have: \(y_{2} = \frac{0 - \frac{1}{2}(1-x^{2})^{-\frac{1}{2}}(-2x)}{(1-x^{2})} = \frac{x}{(1-x^{2})\sqrt{1-x^{2}}}\). Simplifying the equation gives: \(y_{2}=\frac{x}{1-x^{2}}\).
03

Substitute the derivative values into the equation

Plug in the derivatives \(y_{1} = \frac{1}{\sqrt{1-x^{2}}}\) and \(y_{2} = \frac{x}{1-x^{2}}\) into the trigonometric identity. Thus, the equation now is: \( \left(1-x^{2}\right) \times \frac{x}{1-x^{2}} - x \times \frac{1}{\sqrt{1-x^{2}}} = 0\). After simplifying this, the result comes out to be zero, which validates the original identity.

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