Trigonometry Examples

$\frac{\cos (u) \sec (u)}{\tan (u)}$

Step 1

Separate fractions.

$\frac{\sec (u)}{1} \cdot \frac{\cos (u)}{\tan (u)}$

Step 2

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

$\frac{\sec (u)}{1} \cdot \frac{\cos (u)}{\frac{\sin (u)}{\cos (u)}}$

Step 3

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

$\frac{\sec (u)}{1} (\cos (u) \frac{\cos (u)}{\sin (u)})$

Step 4

Write $\cos (u)$ as a fraction with denominator $1$ .

$\frac{\sec (u)}{1} (\frac{\cos (u)}{1} \cdot \frac{\cos (u)}{\sin (u)})$

Step 5

Simplify.

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Step 5.1

Divide $\cos (u)$ by $1$ .

$\frac{\sec (u)}{1} (\cos (u) \frac{\cos (u)}{\sin (u)})$

Step 5.2

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

$\frac{\sec (u)}{1} \cdot \frac{\cos (u) \cos (u)}{\sin (u)}$

Step 6

Divide $\sec (u)$ by $1$ .

$\sec (u) \frac{\cos (u) \cos (u)}{\sin (u)}$

Step 7

Rewrite $\sec (u)$ in terms of sines and cosines.

$\frac{1}{\cos (u)} \cdot \frac{\cos (u) \cos (u)}{\sin (u)}$

Step 8

Simplify the numerator.

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Step 8.1

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

$\frac{1}{\cos (u)} \cdot \frac{\cos^{1} (u) \cos (u)}{\sin (u)}$

Step 8.2

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

$\frac{1}{\cos (u)} \cdot \frac{\cos^{1} (u) \cos^{1} (u)}{\sin (u)}$

Step 8.3

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

$\frac{1}{\cos (u)} \cdot \frac{{\cos (u)}^{1 + 1}}{\sin (u)}$

Step 8.4

Add $1$ and $1$ .

$\frac{1}{\cos (u)} \cdot \frac{\cos^{2} (u)}{\sin (u)}$

Step 9

Simplify terms.

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Step 9.1

Combine.

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

Step 9.2

Cancel the common factor of $\cos^{2} (u)$ and $\cos (u)$ .

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Step 9.2.1

Factor $\cos (u)$ out of $1 \cos^{2} (u)$ .

$\frac{\cos (u) (1 \cos (u))}{\cos (u) \sin (u)}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $\cos (u)$ out of $\cos (u) \sin (u)$ .

$\frac{\cos (u) (1 \cos (u))}{\cos (u) (\sin (u))}$

Step 9.2.2.2

Cancel the common factor.

$\frac{\cos (u) (1 \cos (u))}{\cos (u) \sin (u)}$

Step 9.2.2.3

Rewrite the expression.

$\frac{1 \cos (u)}{\sin (u)}$

Step 9.3

Multiply $\cos (u)$ by $1$ .

$\frac{\cos (u)}{\sin (u)}$

Step 10

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

$\cot (u)$