Calculus Examples

$\cos (B) \cot (B) - \tan (B)$

Step 1

Simplify each term.

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Step 1.1

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

$\cos (B) \frac{\cos (B)}{\sin (B)} - \tan (B)$

Step 1.2

Multiply $\cos (B) \frac{\cos (B)}{\sin (B)}$ .

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Step 1.2.1

Combine $\cos (B)$ and $\frac{\cos (B)}{\sin (B)}$ .

$\frac{\cos (B) \cos (B)}{\sin (B)} - \tan (B)$

Step 1.2.2

Raise $\cos (B)$ to the power of $1$ .

$\frac{\cos^{1} (B) \cos (B)}{\sin (B)} - \tan (B)$

Step 1.2.3

Raise $\cos (B)$ to the power of $1$ .

$\frac{\cos^{1} (B) \cos^{1} (B)}{\sin (B)} - \tan (B)$

Step 1.2.4

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

$\frac{{\cos (B)}^{1 + 1}}{\sin (B)} - \tan (B)$

Step 1.2.5

Add $1$ and $1$ .

$\frac{\cos^{2} (B)}{\sin (B)} - \tan (B)$

Step 1.3

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

$\frac{\cos^{2} (B)}{\sin (B)} - \frac{\sin (B)}{\cos (B)}$

Step 2

To write $\frac{\cos^{2} (B)}{\sin (B)}$ as a fraction with a common denominator, multiply by $\frac{\cos (B)}{\cos (B)}$ .

$\frac{\cos^{2} (B)}{\sin (B)} \cdot \frac{\cos (B)}{\cos (B)} - \frac{\sin (B)}{\cos (B)}$

Step 3

To write $- \frac{\sin (B)}{\cos (B)}$ as a fraction with a common denominator, multiply by $\frac{\sin (B)}{\sin (B)}$ .

$\frac{\cos^{2} (B)}{\sin (B)} \cdot \frac{\cos (B)}{\cos (B)} - \frac{\sin (B)}{\cos (B)} \cdot \frac{\sin (B)}{\sin (B)}$

Step 4

Write each expression with a common denominator of $\sin (B) \cos (B)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{\cos^{2} (B)}{\sin (B)}$ by $\frac{\cos (B)}{\cos (B)}$ .

$\frac{\cos^{2} (B) \cos (B)}{\sin (B) \cos (B)} - \frac{\sin (B)}{\cos (B)} \cdot \frac{\sin (B)}{\sin (B)}$

Step 4.2

Multiply $\frac{\sin (B)}{\cos (B)}$ by $\frac{\sin (B)}{\sin (B)}$ .

$\frac{\cos^{2} (B) \cos (B)}{\sin (B) \cos (B)} - \frac{\sin (B) \sin (B)}{\cos (B) \sin (B)}$

Step 4.3

Reorder the factors of $\sin (B) \cos (B)$ .

$\frac{\cos^{2} (B) \cos (B)}{\cos (B) \sin (B)} - \frac{\sin (B) \sin (B)}{\cos (B) \sin (B)}$

Step 5

Combine the numerators over the common denominator.

$\frac{\cos^{2} (B) \cos (B) - \sin (B) \sin (B)}{\cos (B) \sin (B)}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $\cos^{2} (B)$ by $\cos (B)$ by adding the exponents.

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Step 6.1.1

Multiply $\cos^{2} (B)$ by $\cos (B)$ .

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Step 6.1.1.1

Raise $\cos (B)$ to the power of $1$ .

$\frac{\cos^{2} (B) \cos^{1} (B) - \sin (B) \sin (B)}{\cos (B) \sin (B)}$

Step 6.1.1.2

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

$\frac{{\cos (B)}^{2 + 1} - \sin (B) \sin (B)}{\cos (B) \sin (B)}$

Step 6.1.2

Add $2$ and $1$ .

$\frac{\cos^{3} (B) - \sin (B) \sin (B)}{\cos (B) \sin (B)}$

Step 6.2

Multiply $- \sin (B) \sin (B)$ .

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Step 6.2.1

Raise $\sin (B)$ to the power of $1$ .

$\frac{\cos^{3} (B) - (\sin^{1} (B) \sin (B))}{\cos (B) \sin (B)}$

Step 6.2.2

Raise $\sin (B)$ to the power of $1$ .

$\frac{\cos^{3} (B) - (\sin^{1} (B) \sin^{1} (B))}{\cos (B) \sin (B)}$

Step 6.2.3

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

$\frac{\cos^{3} (B) - {\sin (B)}^{1 + 1}}{\cos (B) \sin (B)}$

Step 6.2.4

Add $1$ and $1$ .

$\frac{\cos^{3} (B) - \sin^{2} (B)}{\cos (B) \sin (B)}$