Trigonometry Examples

$\frac{\tan^{2} (x) - 2 \tan (x) - 3}{\tan (x) + 1}$

Step 1

Simplify the numerator.

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Step 1.1

Let $u = \tan (x)$ . Substitute $u$ for all occurrences of $\tan (x)$ .

$\frac{u^{2} - 2 u - 3}{\tan (x) + 1}$

Step 1.2

Factor $u^{2} - 2 u - 3$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{(u - 3) (u + 1)}{\tan (x) + 1}$

Step 1.3

Replace all occurrences of $u$ with $\tan (x)$ .

$\frac{(\tan (x) - 3) (\tan (x) + 1)}{\tan (x) + 1}$

Step 2

Cancel the common factor of $\tan (x) + 1$ .

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Step 2.1

Cancel the common factor.

$\frac{(\tan (x) - 3) (\tan (x) + 1)}{\tan (x) + 1}$

Step 2.2

Divide $\tan (x) - 3$ by $1$ .

$\tan (x) - 3$