Trigonometry Examples

$\cos (x) (\csc (x) - \tan (x))$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 2.4

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

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Step 2.4.1

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 2.4.2

Cancel the common factor.

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

Step 2.4.3

Rewrite the expression.

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

Step 3

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

$\cot (x) - \sin (x)$