Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Multiply $- - \cot (a)$ .

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Step 2.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2.2

Multiply $\cot (a)$ by $1$ .

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

Step 2.4

Simplify by adding terms.

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Step 2.4.1

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

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Step 2.4.1.1

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

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

Step 2.4.1.2

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

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

Step 2.4.2

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

$2 \tan (a) (2 \cot (a))$

Step 3

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

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Step 3.1

Add parentheses.

$2 (\tan (a) (2 \cot (a)))$

Step 3.2

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

$2 (2 \cot (a) \tan (a))$

Step 3.3

Add parentheses.

$2 (2 (\cot (a) \tan (a)))$

Step 3.4

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

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

Step 3.5

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 $2$ .

$4$