Trigonometry Examples

$\frac{{(1 - \cot (x))}^{2}}{\cot (x)}$

Step 1

Separate fractions.

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 5

Simplify the expression.

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Step 5.1

Divide ${(1 - \cot (x))}^{2}$ by $1$ .

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

Step 5.2

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

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

Step 6

Expand $(1 - \cot (x)) (1 - \cot (x))$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $1$ by $1$ .

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

Step 7.1.2

Multiply $- \cot (x)$ by $1$ .

$(1 - \cot (x) - \cot (x) \cdot 1 - \cot (x) (- \cot (x))) \tan (x)$

Step 7.1.3

Multiply $- 1$ by $1$ .

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

Step 7.1.4

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

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Step 7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.4.2

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

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

Step 7.1.4.3

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

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

Step 7.1.4.4

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

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

Step 7.1.4.5

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

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

Step 7.1.4.6

Add $1$ and $1$ .

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

Step 7.2

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

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

Add parentheses.

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

Step 9.2.2

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

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

Step 9.2.3

Cancel the common factors.

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 9.6

Cancel the common factor of $\cos (x)$ .

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Step 9.6.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 9.6.2

Cancel the common factor.

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

Step 9.6.3

Rewrite the expression.

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

Step 9.7

Cancel the common factor of $\sin (x)$ .

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Step 9.7.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Step 9.7.2

Cancel the common factor.

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

Step 9.7.3

Rewrite the expression.

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

Step 10

Simplify each term.

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Step 10.1

Multiply $- 2$ by $1$ .

$\tan (x) - 2 + \frac{\cos (x)}{\sin (x)}$

Step 10.2

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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