Calculus Examples

$9 \sec^{2} (\frac{π}{4}) + 9 \csc^{2} (\frac{π}{4})$

Step 1

Simplify each term.

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Step 1.1

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

$9 {(\frac{2}{\sqrt{2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.2

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

$9 {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$9 {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$9 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$9 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.5

Add $1$ and $1$ .

$9 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$9 {(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.6.3

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

$9 {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$9 {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.6.4.2

Rewrite the expression.

$9 {(\frac{2 \sqrt{2}}{2^{1}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.3.6.5

Evaluate the exponent.

$9 {(\frac{2 \sqrt{2}}{2})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$9 {(\frac{2 \sqrt{2}}{2})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.4.2

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

$9 {\sqrt{2}}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$9 {(2^{\frac{1}{2}})}^{2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 \cdot 2^{\frac{1}{2} \cdot 2} + 9 \csc^{2} (\frac{π}{4})$

Step 1.5.3

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

$9 \cdot 2^{\frac{2}{2}} + 9 \csc^{2} (\frac{π}{4})$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$9 \cdot 2^{\frac{2}{2}} + 9 \csc^{2} (\frac{π}{4})$

Step 1.5.4.2

Rewrite the expression.

$9 \cdot 2^{1} + 9 \csc^{2} (\frac{π}{4})$

Step 1.5.5

Evaluate the exponent.

$9 \cdot 2 + 9 \csc^{2} (\frac{π}{4})$

Step 1.6

Multiply $9$ by $2$ .

$18 + 9 \csc^{2} (\frac{π}{4})$

Step 1.7

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

$18 + 9 {\sqrt{2}}^{2}$

Step 1.8

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$18 + 9 {(2^{\frac{1}{2}})}^{2}$

Step 1.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$18 + 9 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 1.8.3

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

$18 + 9 \cdot 2^{\frac{2}{2}}$

Step 1.8.4

Cancel the common factor of $2$ .

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Step 1.8.4.1

Cancel the common factor.

$18 + 9 \cdot 2^{\frac{2}{2}}$

Step 1.8.4.2

Rewrite the expression.

$18 + 9 \cdot 2^{1}$

Step 1.8.5

Evaluate the exponent.

$18 + 9 \cdot 2$

Step 1.9

Multiply $9$ by $2$ .

$18 + 18$

Step 2

Add $18$ and $18$ .

$36$