Trigonometry Examples

$\tan^{2} (x) - 3 \sin (x) \tan (x) \sec (x)$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Multiply $- 3 \sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Step 1.4.1

Combine $\frac{\sin (x)}{\cos (x)}$ and $- 3$ .

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

Step 1.4.2

Combine $\frac{\sin (x) \cdot - 3}{\cos (x)}$ and $\sin (x)$ .

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

Step 1.4.3

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{- 3 (\sin^{1} (x) \sin (x))}{\cos (x)} \sec (x)$

Step 1.4.4

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{- 3 (\sin^{1} (x) \sin^{1} (x))}{\cos (x)} \sec (x)$

Step 1.4.5

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{- 3 {\sin (x)}^{1 + 1}}{\cos (x)} \sec (x)$

Step 1.4.6

Add $1$ and $1$ .

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

Step 1.5

Move the negative in front of the fraction.

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

Step 1.6

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{3 \sin^{2} (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$

Step 1.7

Multiply $- \frac{3 \sin^{2} (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 1.7.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{3 \sin^{2} (x)}{\cos (x)}$ .

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

Step 1.7.2

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{3 \sin^{2} (x)}{\cos^{1} (x) \cos (x)}$

Step 1.7.3

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{3 \sin^{2} (x)}{\cos^{1} (x) \cos^{1} (x)}$

Step 1.7.4

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{3 \sin^{2} (x)}{{\cos (x)}^{1 + 1}}$

Step 1.7.5

Add $1$ and $1$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Subtract $3 \sin^{2} (x)$ from $\sin^{2} (x)$ .

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

Step 2.3

Move the negative in front of the fraction.

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

Step 3

Multiply by $1$ .

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

Step 4

Multiply by $1$ .

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

Multiply $2$ by $1$ .

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

Step 8

Divide $2$ by $1$ .

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

Step 9

Multiply $2$ by $- 1$ .

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