Trigonometry Examples

$\sec^{2} (45)$

Step 1

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

${(\frac{2}{\sqrt{2}})}^{2}$

Step 2

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

${(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2}$

Step 3

Combine and simplify the denominator.

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Step 3.1

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

${(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2}$

Step 3.2

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

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2}$

Step 3.3

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

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2}$

Step 3.4

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

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}$

Step 3.5

Add $1$ and $1$ .

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2}$

Step 3.6

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

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Step 3.6.1

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

${(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2}$

Step 3.6.2

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

${(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}$

Step 3.6.3

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

${(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

${(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 3.6.4.2

Rewrite the expression.

${(\frac{2 \sqrt{2}}{2^{1}})}^{2}$

Step 3.6.5

Evaluate the exponent.

${(\frac{2 \sqrt{2}}{2})}^{2}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

${(\frac{2 \sqrt{2}}{2})}^{2}$

Step 4.2

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

${\sqrt{2}}^{2}$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$2^{\frac{1}{2} \cdot 2}$

Step 5.3

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

$2^{\frac{2}{2}}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$2^{\frac{2}{2}}$

Step 5.4.2

Rewrite the expression.

$2^{1}$

Step 5.5

Evaluate the exponent.

$2$