Calculus Examples

$\cos^{2} (\frac{7 π}{8})$

Step 1

The exact value of $\cos (\frac{7 π}{8})$ is $- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 1.1

Rewrite $\frac{7 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos^{2} (\frac{\frac{7 π}{4}}{2})$

Step 1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

${(\pm \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}})}^{2}$

Step 1.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

${(- \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}})}^{2}$

Step 1.4

Simplify $- \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}}$ .

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Step 1.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 1.4.2

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

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

Step 1.4.3

Write $1$ as a fraction with a common denominator.

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

Step 1.4.4

Combine the numerators over the common denominator.

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

Step 1.4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4.6

Multiply $\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.4.6.1

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

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

Step 1.4.6.2

Multiply $2$ by $2$ .

${(- \sqrt{\frac{2 + \sqrt{2}}{4}})}^{2}$

Step 1.4.7

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

${(- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}})}^{2}$

Step 1.4.8

Simplify the denominator.

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Step 1.4.8.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.4.8.2

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

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

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

Apply the product rule to $- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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

Step 2.2

Apply the product rule to $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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

Step 3

Simplify the expression.

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Step 3.1

Raise $- 1$ to the power of $2$ .

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

Step 3.2

Multiply $\frac{{\sqrt{2 + \sqrt{2}}}^{2}}{2^{2}}$ by $1$ .

$\frac{{\sqrt{2 + \sqrt{2}}}^{2}}{2^{2}}$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Simplify.

$\frac{2 + \sqrt{2}}{2^{2}}$

Step 5

Raise $2$ to the power of $2$ .

$\frac{2 + \sqrt{2}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2 + \sqrt{2}}{4}$

Decimal Form:

$0.85355339 \dots$