Precalculus Examples

$1 - 2 \sin^{2} (67.5)$

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

Tap for more steps...

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

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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 $2$ .

Tap for more steps...

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:

$- 0.70710678 \dots$