Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

$\tan^{1} (x) \tan (x) + \tan (x) \cot (x) + \cot (x) \tan (x) + \cot (x) \cot (x)$

Step 3.1.1.2

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

$\tan^{1} (x) \tan^{1} (x) + \tan (x) \cot (x) + \cot (x) \tan (x) + \cot (x) \cot (x)$

Step 3.1.1.3

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

${\tan (x)}^{1 + 1} + \tan (x) \cot (x) + \cot (x) \tan (x) + \cot (x) \cot (x)$

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

$\tan^{2} (x) + \frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cot (x) \tan (x) + \cot (x) \cot (x)$

Step 3.1.2.3

Cancel the common factors.

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

Step 3.1.3

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

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Step 3.1.3.1

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

$\tan^{2} (x) + 1 + \frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cot (x) \cot (x)$

Step 3.1.3.2

Cancel the common factors.

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

Step 3.1.4

Multiply $\cot (x) \cot (x)$ .

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Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

Add $1$ and $1$ .

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

Step 3.2

Add $1$ and $1$ .

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