Precalculus Examples

$\tan (\frac{7 π}{4} + x)$

Step 1

Apply the sum of angles identity.

$\frac{\tan (\frac{7 π}{4}) + \tan (x)}{1 - \tan (\frac{7 π}{4}) \tan (x)}$

Step 2

Simplify the numerator.

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Step 2.1

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 fourth quadrant.

$\frac{- \tan (\frac{π}{4}) + \tan (x)}{1 - \tan (\frac{7 π}{4}) \tan (x)}$

Step 2.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{- 1 \cdot 1 + \tan (x)}{1 - \tan (\frac{7 π}{4}) \tan (x)}$

Step 2.3

Multiply $- 1$ by $1$ .

$\frac{- 1 + \tan (x)}{1 - \tan (\frac{7 π}{4}) \tan (x)}$

Step 3

Simplify the denominator.

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Step 3.1

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 fourth quadrant.

$\frac{- 1 + \tan (x)}{1 - - \tan (\frac{π}{4}) \tan (x)}$

Step 3.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{- 1 + \tan (x)}{1 - (- 1 \cdot 1) \tan (x)}$

Step 3.3

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

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Step 3.3.1

Multiply $- 1$ by $1$ .

$\frac{- 1 + \tan (x)}{1 - - 1 \tan (x)}$

Step 3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{- 1 + \tan (x)}{1 + 1 \tan (x)}$

Step 3.4

Multiply $\tan (x)$ by $1$ .

$\frac{- 1 + \tan (x)}{1 + \tan (x)}$

Step 4

Simplify with factoring out.

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Step 4.1

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) + \tan (x)}{1 + \tan (x)}$

Step 4.2

Factor $- 1$ out of $\tan (x)$ .

$\frac{- 1 (1) - 1 (- \tan (x))}{1 + \tan (x)}$

Step 4.3

Factor $- 1$ out of $- 1 (1) - 1 (- \tan (x))$ .

$\frac{- 1 (1 - \tan (x))}{1 + \tan (x)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{1 - \tan (x)}{1 + \tan (x)}$