Algebra Examples

$\sec (x) (\csc (x) - 2 \sin (x)) = \cot (x) - \tan (x)$

Step 1

Start on the right side.

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

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.

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

Step 2.2

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

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

Step 3

Subtract fractions.

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Step 3.1

To write $\frac{\cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 3.2

To write $- \frac{\sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.3

Write each expression with a common denominator of $\sin (x) \cos (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Reorder the factors of $\sin (x) \cos (x)$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

Subtract ${\sin (x)}^{2}$ from $- {\sin (x)}^{2}$ .

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

Step 7

Now consider the left side of the equation.

$\sec (x) (\csc (x) - 2 \sin (x))$

Step 8

Convert to sines and cosines.

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Step 8.1

Apply the reciprocal identity to $\sec (x)$ .

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

Step 8.2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

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

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

Step 9.3

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

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Step 9.3.1

Combine $- 2$ and $\frac{1}{\cos (x)}$ .

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

Step 9.3.2

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

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

Step 9.4

Move the negative in front of the fraction.

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

Step 10

Subtract fractions.

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Step 10.1

To write $- \frac{2 \sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 10.2

Multiply $\frac{2 \sin (x)}{\cos (x)}$ by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 10.3

Combine the numerators over the common denominator.

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

Step 11

Multiply $- 2 \sin (x) \sin (x)$ .

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

Step 12

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

$\sec (x) (\csc (x) - 2 \sin (x)) = \cot (x) - \tan (x)$ is an identity