Calculus Examples

$lim_{x \to \frac{π}{2}} \frac{\cos (\frac{x}{2}) - \sin (\frac{x}{2})}{\cos (x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \frac{π}{2}} \cos (\frac{x}{2}) - \sin (\frac{x}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$\frac{lim_{x \to \frac{π}{2}} \cos (\frac{x}{2}) - lim_{x \to \frac{π}{2}} \sin (\frac{x}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.2

Move the limit inside the trig function because cosine is continuous.

$\frac{\cos (lim_{x \to \frac{π}{2}} \frac{x}{2}) - lim_{x \to \frac{π}{2}} \sin (\frac{x}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.3

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\cos (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - lim_{x \to \frac{π}{2}} \sin (\frac{x}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.4

Move the limit inside the trig function because sine is continuous.

$\frac{\cos (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - \sin (lim_{x \to \frac{π}{2}} \frac{x}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.5

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\cos (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - \sin (\frac{1}{2} lim_{x \to \frac{π}{2}} x)}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.6

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $x$ .

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Step 1.1.2.6.1

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{\cos (\frac{1}{2} \cdot \frac{π}{2}) - \sin (\frac{1}{2} lim_{x \to \frac{π}{2}} x)}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.6.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{\cos (\frac{1}{2} \cdot \frac{π}{2}) - \sin (\frac{1}{2} \cdot \frac{π}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

Multiply $\frac{1}{2} \cdot \frac{π}{2}$ .

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Step 1.1.2.7.1.1.1

Multiply $\frac{1}{2}$ by $\frac{π}{2}$ .

$\frac{\cos (\frac{π}{2 \cdot 2}) - \sin (\frac{1}{2} \cdot \frac{π}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.1.1.2

Multiply $2$ by $2$ .

$\frac{\cos (\frac{π}{4}) - \sin (\frac{1}{2} \cdot \frac{π}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} - \sin (\frac{1}{2} \cdot \frac{π}{2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.1.3

Multiply $\frac{1}{2} \cdot \frac{π}{2}$ .

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Step 1.1.2.7.1.3.1

Multiply $\frac{1}{2}$ by $\frac{π}{2}$ .

$\frac{\frac{\sqrt{2}}{2} - \sin (\frac{π}{2 \cdot 2})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.1.3.2

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{2}}{2} - \sin (\frac{π}{4})}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.1.4

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.2

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2} - \sqrt{2}}{2}}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.3

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{\frac{0}{2}}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.2.7.4

Divide $0$ by $2$ .

$\frac{0}{lim_{x \to \frac{π}{2}} \cos (x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Move the limit inside the trig function because cosine is continuous.

$\frac{0}{\cos (lim_{x \to \frac{π}{2}} x)}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{0}{\cos (\frac{π}{2})}$

Step 1.1.3.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \frac{π}{2}} \frac{\cos (\frac{x}{2}) - \sin (\frac{x}{2})}{\cos (x)} = lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (\frac{x}{2}) - \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (\frac{x}{2}) - \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.2

By the Sum Rule, the derivative of $\cos (\frac{x}{2}) - \sin (\frac{x}{2})$ with respect to $x$ is $\frac{d}{d x} [\cos (\frac{x}{2})] + \frac{d}{d x} [- \sin (\frac{x}{2})]$ .

$lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (\frac{x}{2})] + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\cos (\frac{x}{2})]$ .

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Step 1.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{x}{2}$ .

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Step 1.3.3.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{x}{2}$ .

$lim_{x \to \frac{π}{2}} \frac{\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.3.1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$lim_{x \to \frac{π}{2}} \frac{- \sin (u_{1}) \frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.3.1.3

Replace all occurrences of $u_{1}$ with $\frac{x}{2}$ .

$lim_{x \to \frac{π}{2}} \frac{- \sin (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$lim_{x \to \frac{π}{2}} \frac{- \sin (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x]) + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \frac{π}{2}} \frac{- \sin (\frac{x}{2}) (\frac{1}{2} \cdot 1) + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.3.4

Multiply $\frac{1}{2}$ by $1$ .

$lim_{x \to \frac{π}{2}} \frac{- \sin (\frac{x}{2}) \frac{1}{2} + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.3.5

Combine $\frac{1}{2}$ and $\sin (\frac{x}{2})$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} + \frac{d}{d x} [- \sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \sin (\frac{x}{2})]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (\frac{x}{2})$ with respect to $x$ is $- \frac{d}{d x} [\sin (\frac{x}{2})]$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - \frac{d}{d x} [\sin (\frac{x}{2})]}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{x}{2}$ .

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Step 1.3.4.2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{2}$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [\frac{x}{2}])}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - (\cos (u_{2}) \frac{d}{d x} [\frac{x}{2}])}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4.2.3

Replace all occurrences of $u_{2}$ with $\frac{x}{2}$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - (\cos (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}])}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - (\cos (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x]))}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - (\cos (\frac{x}{2}) (\frac{1}{2} \cdot 1))}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4.5

Multiply $\frac{1}{2}$ by $1$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - (\cos (\frac{x}{2}) \frac{1}{2})}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.4.6

Combine $\cos (\frac{x}{2})$ and $\frac{1}{2}$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - \frac{\cos (\frac{x}{2})}{2}}{\frac{d}{d x} [\cos (x)]}$

Step 1.3.5

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to \frac{π}{2}} \frac{- \frac{\sin (\frac{x}{2})}{2} - \frac{\cos (\frac{x}{2})}{2}}{- \sin (x)}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to \frac{π}{2}} \frac{\frac{- \sin (\frac{x}{2}) - \cos (\frac{x}{2})}{2}}{- \sin (x)}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$\frac{lim_{x \to \frac{π}{2}} \frac{- \sin (\frac{x}{2}) - \cos (\frac{x}{2})}{2}}{lim_{x \to \frac{π}{2}} - \sin (x)}$

Step 2.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} lim_{x \to \frac{π}{2}} - \sin (\frac{x}{2}) - \cos (\frac{x}{2})}{lim_{x \to \frac{π}{2}} - \sin (x)}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$\frac{\frac{1}{2} (- lim_{x \to \frac{π}{2}} \sin (\frac{x}{2}) - lim_{x \to \frac{π}{2}} \cos (\frac{x}{2}))}{lim_{x \to \frac{π}{2}} - \sin (x)}$

Step 2.4

Move the limit inside the trig function because sine is continuous.

$\frac{\frac{1}{2} (- \sin (lim_{x \to \frac{π}{2}} \frac{x}{2}) - lim_{x \to \frac{π}{2}} \cos (\frac{x}{2}))}{lim_{x \to \frac{π}{2}} - \sin (x)}$

Step 2.5

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - lim_{x \to \frac{π}{2}} \cos (\frac{x}{2}))}{lim_{x \to \frac{π}{2}} - \sin (x)}$

Step 2.6

Move the limit inside the trig function because cosine is continuous.

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - \cos (lim_{x \to \frac{π}{2}} \frac{x}{2}))}{lim_{x \to \frac{π}{2}} - \sin (x)}$

Step 2.7

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - \cos (\frac{1}{2} lim_{x \to \frac{π}{2}} x))}{lim_{x \to \frac{π}{2}} - \sin (x)}$

Step 2.8

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - \cos (\frac{1}{2} lim_{x \to \frac{π}{2}} x))}{- lim_{x \to \frac{π}{2}} \sin (x)}$

Step 2.9

Move the limit inside the trig function because sine is continuous.

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} lim_{x \to \frac{π}{2}} x) - \cos (\frac{1}{2} lim_{x \to \frac{π}{2}} x))}{- \sin (lim_{x \to \frac{π}{2}} x)}$

Step 3

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} \cdot \frac{π}{2}) - \cos (\frac{1}{2} lim_{x \to \frac{π}{2}} x))}{- \sin (lim_{x \to \frac{π}{2}} x)}$

Step 3.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} \cdot \frac{π}{2}) - \cos (\frac{1}{2} \cdot \frac{π}{2}))}{- \sin (lim_{x \to \frac{π}{2}} x)}$

Step 3.3

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{\frac{1}{2} (- \sin (\frac{1}{2} \cdot \frac{π}{2}) - \cos (\frac{1}{2} \cdot \frac{π}{2}))}{- \sin (\frac{π}{2})}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $- 1$ out of $- \sin (\frac{1}{2} \cdot \frac{π}{2}) - \cos (\frac{1}{2} \cdot \frac{π}{2})$ .

$\frac{\frac{1}{2} (- (\sin (\frac{1}{2} \cdot \frac{π}{2}) + \cos (\frac{1}{2} \cdot \frac{π}{2})))}{- \sin (\frac{π}{2})}$

Step 4.1.2

Multiply $\frac{1}{2} \cdot \frac{π}{2}$ .

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Step 4.1.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{2}$ .

$\frac{\frac{1}{2} (- (\sin (\frac{π}{2 \cdot 2}) + \cos (\frac{1}{2} \cdot \frac{π}{2})))}{- \sin (\frac{π}{2})}$

Step 4.1.2.2

Multiply $2$ by $2$ .

$\frac{\frac{1}{2} (- (\sin (\frac{π}{4}) + \cos (\frac{1}{2} \cdot \frac{π}{2})))}{- \sin (\frac{π}{2})}$

Step 4.1.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{1}{2} (- (\frac{\sqrt{2}}{2} + \cos (\frac{1}{2} \cdot \frac{π}{2})))}{- \sin (\frac{π}{2})}$

Step 4.1.4

Multiply $\frac{1}{2} \cdot \frac{π}{2}$ .

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Step 4.1.4.1

Multiply $\frac{1}{2}$ by $\frac{π}{2}$ .

$\frac{\frac{1}{2} (- (\frac{\sqrt{2}}{2} + \cos (\frac{π}{2 \cdot 2})))}{- \sin (\frac{π}{2})}$

Step 4.1.4.2

Multiply $2$ by $2$ .

$\frac{\frac{1}{2} (- (\frac{\sqrt{2}}{2} + \cos (\frac{π}{4})))}{- \sin (\frac{π}{2})}$

Step 4.1.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{1}{2} (- (\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}))}{- \sin (\frac{π}{2})}$

Step 4.1.6

Combine the numerators over the common denominator.

$\frac{\frac{1}{2} (- \frac{\sqrt{2} + \sqrt{2}}{2})}{- \sin (\frac{π}{2})}$

Step 4.1.7

Rewrite $\frac{\sqrt{2} + \sqrt{2}}{2}$ in a factored form.

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Step 4.1.7.1

Add $\sqrt{2}$ and $\sqrt{2}$ .

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

Step 4.1.7.2

Reduce the expression by cancelling the common factors.

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Step 4.1.7.2.1

Reduce the expression $\frac{2 \sqrt{2}}{2}$ by cancelling the common factors.

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Step 4.1.7.2.1.1

Cancel the common factor.

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

Step 4.1.7.2.1.2

Rewrite the expression.

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

Step 4.1.7.2.2

Divide $\sqrt{2}$ by $1$ .

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

$\frac{\frac{1}{2} \cdot - 1 \sqrt{2}}{- \sin (\frac{π}{2})}$

Step 4.1.8

Combine exponents.

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Step 4.1.8.1

Factor out negative.

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

Step 4.1.8.2

Combine $\frac{1}{2}$ and $\sqrt{2}$ .

$\frac{- \frac{\sqrt{2}}{2}}{- \sin (\frac{π}{2})}$

Step 4.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{- \frac{\sqrt{2}}{2}}{- 1 \cdot 1}$

Step 4.3

Multiply $- 1$ by $1$ .

$\frac{- \frac{\sqrt{2}}{2}}{- 1}$

Step 4.4

Dividing two negative values results in a positive value.

$\frac{\frac{\sqrt{2}}{2}}{1}$

Step 4.5

Divide $\frac{\sqrt{2}}{2}$ by $1$ .

$\frac{\sqrt{2}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$