Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

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

$\cot (x) \cdot (\sec (x) - \frac{\sin (x)}{\cos (x)})$

Step 2.2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Add parentheses.

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Cancel the common factors.

$\frac{1}{\sin (x)} - 1 \cdot 1$

Step 6

Simplify each term.

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Step 6.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x) - 1 \cdot 1$

Step 6.2

Multiply $- 1$ by $1$ .

$\csc (x) - 1$