Trigonometry Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cot (x) + \tan (x)$ and $b = \cot (x) - \tan (x)$ .

$(\cot (x) + \tan (x) + \cot (x) - \tan (x)) (\cot (x) + \tan (x) - (\cot (x) - \tan (x)))$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\cot (x) + \tan (x) + \cot (x) - \tan (x)$ .

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Step 2.1.1

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

$(\cot (x) + 0 + \cot (x)) (\cot (x) + \tan (x) - (\cot (x) - \tan (x)))$

Step 2.1.2

Add $\cot (x)$ and $0$ .

$(\cot (x) + \cot (x)) (\cot (x) + \tan (x) - (\cot (x) - \tan (x)))$

Step 2.2

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

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

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Multiply $- - \tan (x)$ .

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Step 2.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2.2

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

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

Step 2.4

Simplify by adding terms.

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Step 2.4.1

Combine the opposite terms in $\cot (x) + \tan (x) - \cot (x) + \tan (x)$ .

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Step 2.4.1.1

Subtract $\cot (x)$ from $\cot (x)$ .

$2 \cot (x) (0 + \tan (x) + \tan (x))$

Step 2.4.1.2

Add $0$ and $\tan (x)$ .

$2 \cot (x) (\tan (x) + \tan (x))$

Step 2.4.2

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

$2 \cot (x) (2 \tan (x))$

Step 3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.1

Add parentheses.

$2 (\cot (x) (2 \tan (x)))$

Step 3.2

Reorder $\cot (x)$ and $2 \tan (x)$ .

$2 (2 \tan (x) \cot (x))$

Step 3.3

Add parentheses.

$2 (2 (\tan (x) \cot (x)))$

Step 3.4

Reorder $\tan (x)$ and $\cot (x)$ .

$2 (2 (\cot (x) \tan (x)))$

Step 3.5

Rewrite $2 \cot (x) (2 \tan (x))$ in terms of sines and cosines.

$2 (2 (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)}))$

Step 3.6

Cancel the common factors.

$2 (2 \cdot 1)$

Step 4

Multiply $2 (2 \cdot 1)$ .

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Step 4.1

Multiply $2$ by $1$ .

$2 \cdot 2$

Step 4.2

Multiply $2$ by $2$ .

$4$