Trigonometry Examples

$\frac{2 \sin (x) \cot (x) + \sin (x) - 4 \cot (x) - 2}{2 \cot (x) + 1} = \sin (x) - 2$

Step 1

Start on the left side.

$\frac{2 \sin (x) \cot (x) + \sin (x) - 4 \cot (x) - 2}{2 \cot (x) + 1}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{2 \sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x) - 4 \cot (x) - 2}{2 \cot (x) + 1}$

Step 2.2

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

$\frac{2 \sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x) - 4 \frac{\cos (x)}{\sin (x)} - 2}{2 \cot (x) + 1}$

Step 2.3

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

$\frac{2 \sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x) - 4 \frac{\cos (x)}{\sin (x)} - 2}{2 \frac{\cos (x)}{\sin (x)} + 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor out the greatest common factor from each group.

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Step 3.1.1.1

Group the first two terms and the last two terms.

$\frac{(2 \sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x)) - 4 \frac{\cos (x)}{\sin (x)} - 2}{2 \frac{\cos (x)}{\sin (x)} + 1}$

Step 3.1.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.1.2

Factor the polynomial by factoring out the greatest common factor, $2 \frac{\cos (x)}{\sin (x)} + 1$ .

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

Step 3.1.3

Combine $2$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 3.1.4

Write $1$ as a fraction with a common denominator.

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

Step 3.1.5

Combine the numerators over the common denominator.

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Combine $2$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.3

Cancel the common factor of $\frac{2 \cos (x) + \sin (x)}{\sin (x)}$ .

$\sin (x) - 2$

Step 4

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

$\frac{2 \sin (x) \cot (x) + \sin (x) - 4 \cot (x) - 2}{2 \cot (x) + 1} = \sin (x) - 2$ is an identity