Trigonometry Examples

$\cos^{2} (67.5) - \sin^{2} (67.5)$

Step 1

Apply the cosine double-angle identity.

$\cos (2 \cdot 67.5)$

Step 2

Multiply $2$ by $67.5$ .

$\cos (135)$

Step 3

The exact value of $\cos (135)$ is $- \frac{\sqrt{2}}{2}$ .

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Step 3.1

Rewrite $135$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{270}{2})$

Step 3.2

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

$\pm \sqrt{\frac{1 + \cos (270)}{2}}$

Step 3.3

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

$- \sqrt{\frac{1 + \cos (270)}{2}}$

Step 3.4

Simplify $- \sqrt{\frac{1 + \cos (270)}{2}}$ .

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Step 3.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$- \sqrt{\frac{1 - \cos (90)}{2}}$

Step 3.4.2

The exact value of $\cos (90)$ is $0$ .

$- \sqrt{\frac{1 - 0}{2}}$

Step 3.4.3

Multiply $- 1$ by $0$ .

$- \sqrt{\frac{1 + 0}{2}}$

Step 3.4.4

Add $1$ and $0$ .

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

Step 3.4.5

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 3.4.6

Any root of $1$ is $1$ .

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

Step 3.4.7

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

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

Step 3.4.8

Combine and simplify the denominator.

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Step 3.4.8.1

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

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

Step 3.4.8.2

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

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

Step 3.4.8.3

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

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

Step 3.4.8.4

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

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

Step 3.4.8.5

Add $1$ and $1$ .

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

Step 3.4.8.6

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

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Step 3.4.8.6.1

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

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

Step 3.4.8.6.2

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

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

Step 3.4.8.6.3

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

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

Step 3.4.8.6.4

Cancel the common factor of $2$ .

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Step 3.4.8.6.4.1

Cancel the common factor.

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

Step 3.4.8.6.4.2

Rewrite the expression.

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

Step 3.4.8.6.5

Evaluate the exponent.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.70710678 \dots$