Trigonometry Examples

$\cos^{2} (\frac{π}{6}) - \sin^{2} (\frac{π}{2})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

${(\frac{\sqrt{3}}{2})}^{2} - \sin^{2} (\frac{π}{2})$

Step 1.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{{\sqrt{3}}^{2}}{2^{2}} - \sin^{2} (\frac{π}{2})$

Step 1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} - \sin^{2} (\frac{π}{2})$

Step 1.3.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} - \sin^{2} (\frac{π}{2})$

Step 1.3.3

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

$\frac{3^{\frac{2}{2}}}{2^{2}} - \sin^{2} (\frac{π}{2})$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2^{2}} - \sin^{2} (\frac{π}{2})$

Step 1.3.4.2

Rewrite the expression.

$\frac{3^{1}}{2^{2}} - \sin^{2} (\frac{π}{2})$

Step 1.3.5

Evaluate the exponent.

$\frac{3}{2^{2}} - \sin^{2} (\frac{π}{2})$

Step 1.4

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

$\frac{3}{4} - \sin^{2} (\frac{π}{2})$

Step 1.5

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{3}{4} - 1^{2}$

Step 1.6

One to any power is one.

$\frac{3}{4} - 1 \cdot 1$

Step 1.7

Multiply $- 1$ by $1$ .

$\frac{3}{4} - 1$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3}{4} - 1 \cdot \frac{4}{4}$

Step 3

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{3}{4} + \frac{- 1 \cdot 4}{4}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 - 1 \cdot 4}{4}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $4$ .

$\frac{3 - 4}{4}$

Step 5.2

Subtract $4$ from $3$ .

$\frac{- 1}{4}$

Step 6

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$