Trigonometry Examples

$\sin (x) \tan^{2} (x) - \sin (x)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.3.2.1

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

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Step 1.3.2.1.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3.2.2

Add $1$ and $2$ .

$\frac{\sin^{3} (x)}{\cos^{2} (x)} - \sin (x)$

Step 2

Simplify each term.

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Step 2.1

Factor $\sin^{2} (x)$ out of $\sin^{3} (x)$ .

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

Step 2.2

Multiply by $1$ .

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

Step 2.3

Separate fractions.

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

Step 2.4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 2.5

Divide $\sin (x)$ by $1$ .

$\sin (x) \tan^{2} (x) - \sin (x)$