Algebra Examples

$\cot (x) - \cos (x) = \cot (x) (1 - \sin (x))$

Step 1

Start on the right side.

$\cot (x) (1 - \sin (x))$

Step 2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Simplify each term.

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

Step 4

Now consider the left side of the equation.

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

Step 5

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cot (x) - \cos (x) = \cot (x) (1 - \sin (x))$ is an identity