Trigonometry Examples

$\frac{1 + \csc (x)}{\cos (x) \cot (x)}$

Step 1

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

$\frac{1 + \frac{1}{\sin (x)}}{\cos (x) \cot (x)}$

Step 2

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

$\frac{1 + \frac{1}{\sin (x)}}{\cos (x) \frac{\cos (x)}{\sin (x)}}$

Step 3

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

$\frac{1 + \frac{1}{\sin (x)}}{\frac{\cos (x) \cos (x)}{\sin (x)}}$

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Simplify terms.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $\frac{\sin (x)}{\cos^{2} (x)}$ by $1$ .

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

Step 6.3

Combine.

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

Step 6.4

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

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

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

Step 7

Simplify each term.

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Step 7.1

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

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

Step 7.2

Separate fractions.

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

Step 7.3

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

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

Step 7.4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x) + \frac{1}{\cos^{2} (x)}$

Step 7.5

Rewrite $1$ as $1^{2}$ .

$\sec (x) \tan (x) + \frac{1^{2}}{\cos^{2} (x)}$

Step 7.6

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$\sec (x) \tan (x) + {(\frac{1}{\cos (x)})}^{2}$

Step 7.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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