Trigonometry Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite ${(1 + \tan (x))}^{2}$ as $(1 + \tan (x)) (1 + \tan (x))$ .

$2 \tan (x) - ((1 + \tan (x)) (1 + \tan (x)))$

Step 1.2

Expand $(1 + \tan (x)) (1 + \tan (x))$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$2 \tan (x) - (1 (1 + \tan (x)) + \tan (x) (1 + \tan (x)))$

Step 1.2.2

Apply the distributive property.

$2 \tan (x) - (1 \cdot 1 + 1 \tan (x) + \tan (x) (1 + \tan (x)))$

Step 1.2.3

Apply the distributive property.

$2 \tan (x) - (1 \cdot 1 + 1 \tan (x) + \tan (x) \cdot 1 + \tan (x) \tan (x))$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $1$ by $1$ .

$2 \tan (x) - (1 + 1 \tan (x) + \tan (x) \cdot 1 + \tan (x) \tan (x))$

Step 1.3.1.2

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

$2 \tan (x) - (1 + \tan (x) + \tan (x) \cdot 1 + \tan (x) \tan (x))$

Step 1.3.1.3

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

$2 \tan (x) - (1 + \tan (x) + \tan (x) + \tan (x) \tan (x))$

Step 1.3.1.4

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

Tap for more steps...

Step 1.3.1.4.1

Raise $\tan (x)$ to the power of $1$ .

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

Step 1.3.1.4.2

Raise $\tan (x)$ to the power of $1$ .

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

Step 1.3.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3.1.4.4

Add $1$ and $1$ .

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

Step 1.3.2

Add $\tan (x)$ and $\tan (x)$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Multiply $- 1$ by $1$ .

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

Step 1.5.2

Multiply $2$ by $- 1$ .

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Combine the opposite terms in $2 \tan (x) - 1 - 2 \tan (x) - \tan^{2} (x)$ .

Tap for more steps...

Step 2.1.1

Subtract $2 \tan (x)$ from $2 \tan (x)$ .

$0 - 1 - \tan^{2} (x)$

Step 2.1.2

Subtract $1$ from $0$ .

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

Step 2.2

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

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

Step 2.3

Factor $- 1$ out of $- \tan^{2} (x)$ .

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

Step 2.4

Factor $- 1$ out of $- 1 (1) - (\tan^{2} (x))$ .

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

Step 2.5

Rewrite $- 1 (1 + \tan^{2} (x))$ as $- (1 + \tan^{2} (x))$ .

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

Step 3

Rearrange terms.

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

Step 4

Apply pythagorean identity.

$- \sec^{2} (x)$