Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = \cos (x)$ .

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

Step 3

Reorder terms.

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

Step 4

Now consider the left side of the equation.

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

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Subtract fractions.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 7

Simplify each term.

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

Step 8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = \cos (x)$ .

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

Step 9

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

$\tan (x) - \cot (x) = \frac{\sin^{2} (x) - \cos^{2} (x)}{\sin (x) \cos (x)}$ is an identity