Trigonometry Examples

$\cos (x) (\csc (x) - \sec (x)) - \cot (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

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

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

Step 1.4

Rewrite using the commutative property of multiplication.

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

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

Step 1.6

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

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

Step 2

Combine the opposite terms in $\frac{\cos (x)}{\sin (x)} - 1 - \frac{\cos (x)}{\sin (x)}$ .

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Step 2.1

Subtract $\frac{\cos (x)}{\sin (x)}$ from $\frac{\cos (x)}{\sin (x)}$ .

$0 - 1$

Step 2.2

Subtract $1$ from $0$ .

$- 1$