Precalculus Examples

$2 \sin^{2} (225) - \tan (135)$

Step 1

Simplify each term.

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Step 1.1

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

$2 {(- \sin (45))}^{2} - \tan (135)$

Step 1.2

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

$2 {(- \frac{\sqrt{2}}{2})}^{2} - \tan (135)$

Step 1.3

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

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Step 1.3.1

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

$2 ({(- 1)}^{2} {(\frac{\sqrt{2}}{2})}^{2}) - \tan (135)$

Step 1.3.2

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

$2 ({(- 1)}^{2} \frac{{\sqrt{2}}^{2}}{2^{2}}) - \tan (135)$

Step 1.4

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

$2 (1 \frac{{\sqrt{2}}^{2}}{2^{2}}) - \tan (135)$

Step 1.5

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

$2 \frac{{\sqrt{2}}^{2}}{2^{2}} - \tan (135)$

Step 1.6

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

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Step 1.6.1

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

$2 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} - \tan (135)$

Step 1.6.2

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

$2 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - \tan (135)$

Step 1.6.3

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

$2 \frac{2^{\frac{2}{2}}}{2^{2}} - \tan (135)$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$2 \frac{2^{\frac{2}{2}}}{2^{2}} - \tan (135)$

Step 1.6.4.2

Rewrite the expression.

$2 \frac{2^{1}}{2^{2}} - \tan (135)$

Step 1.6.5

Evaluate the exponent.

$2 \frac{2}{2^{2}} - \tan (135)$

Step 1.7

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

$2 (\frac{2}{4}) - \tan (135)$

Step 1.8

Cancel the common factor of $2$ .

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Step 1.8.1

Factor $2$ out of $4$ .

$2 \frac{2}{2 (2)} - \tan (135)$

Step 1.8.2

Cancel the common factor.

$2 \frac{2}{2 \cdot 2} - \tan (135)$

Step 1.8.3

Rewrite the expression.

$\frac{2}{2} - \tan (135)$

Step 1.9

Divide $2$ by $2$ .

$1 - \tan (135)$

Step 1.10

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

$1 - - \tan (45)$

Step 1.11

The exact value of $\tan (45)$ is $1$ .

$1 - (- 1 \cdot 1)$

Step 1.12

Multiply $- (- 1 \cdot 1)$ .

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Step 1.12.1

Multiply $- 1$ by $1$ .

$1 - - 1$

Step 1.12.2

Multiply $- 1$ by $- 1$ .

$1 + 1$

Step 2

Add $1$ and $1$ .

$2$