Trigonometry Examples

$\sec^{2} (x) \cot (x) - \cot (x) = \tan (x)$

Step 1

Start on the left side.

$\sec^{2} (x) \cot (x) - \cot (x)$

Step 2

Apply Pythagorean identity in reverse.

$(\tan^{2} (x) + 1) \cot (x) - \cot (x)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Combine.

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

Step 4.1.3

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

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

Step 4.1.4

Simplify each term.

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Step 4.1.4.1

Cancel the common factor of ${\sin (x)}^{2}$ and $\sin (x)$ .

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Step 4.1.4.1.1

Factor $\sin (x)$ out of ${\sin (x)}^{2} \cos (x)$ .

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

Step 4.1.4.1.2

Cancel the common factors.

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Step 4.1.4.1.2.1

Factor $\sin (x)$ out of ${\cos (x)}^{2} \sin (x)$ .

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

Step 4.1.4.1.2.2

Cancel the common factor.

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

Step 4.1.4.1.2.3

Rewrite the expression.

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

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $\cos (x)$ out of ${\cos (x)}^{2}$ .

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

Step 4.1.4.2.2

Cancel the common factor.

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

Step 4.1.4.2.3

Rewrite the expression.

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Subtract $\cos (x)$ from $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{0}{\sin (x)}$

Step 4.4

Divide $0$ by $\sin (x)$ .

$\frac{\sin (x)}{\cos (x)} + 0$

Step 4.5

Add $\frac{\sin (x)}{\cos (x)}$ and $0$ .

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

Step 5

Rewrite $\frac{\sin (x)}{\cos (x)}$ as $\tan (x)$ .

$\tan (x)$

Step 6

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

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