Calculus Examples

\frac{{(1 - x^{2})}^{- \frac{1}{2}} sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sqrt{1 - {cos}^{2} ({(1 - x^{2})}^{\frac{1}{2}})}}

$\frac{{(1 - x^{2})}^{- \frac{1}{2}} \sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sqrt{1 - \cos^{2} ({(1 - x^{2})}^{\frac{1}{2}})}}$

Step 1

Apply pythagorean identity.

\frac{{(1 - x^{2})}^{- \frac{1}{2}} sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sqrt{{sin}^{2} ({(1 - x^{2})}^{\frac{1}{2}})}}

$\frac{{(1 - x^{2})}^{- \frac{1}{2}} \sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sqrt{\sin^{2} ({(1 - x^{2})}^{\frac{1}{2}})}}$

Step 2

Move

{(1 - x^{2})}^{- \frac{1}{2}}

${(1 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\frac{sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sqrt{{sin}^{2} ({(1 - x^{2})}^{\frac{1}{2}})} {(1 - x^{2})}^{\frac{1}{2}}}

$\frac{\sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sqrt{\sin^{2} ({(1 - x^{2})}^{\frac{1}{2}})} {(1 - x^{2})}^{\frac{1}{2}}}$

Step 3

Pull terms out from under the radical, assuming positive real numbers.

\frac{sin ({(1 - x^{2})}^{\frac{1}{2}})}{sin ({(1 - x^{2})}^{\frac{1}{2}}) {(1 - x^{2})}^{\frac{1}{2}}}

$\frac{\sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sin ({(1 - x^{2})}^{\frac{1}{2}}) {(1 - x^{2})}^{\frac{1}{2}}}$

Step 4

Cancel the common factor.

$\frac{\sin ({(1 - x^{2})}^{\frac{1}{2}})}{\sin ({(1 - x^{2})}^{\frac{1}{2}}) {(1 - x^{2})}^{\frac{1}{2}}}$

Step 5

Rewrite the expression.

$\frac{1}{{(1 - x^{2})}^{\frac{1}{2}}}$