Trigonometry Examples

$\tan (x) \sin (x) + \sec (x) \cos^{2} (x)$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $\cos (x)$ .

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

Separate fractions.

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

Step 2.3

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

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

Step 2.4

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

$\sin (x) \tan (x) + \cos (x)$