Trigonometry Examples

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

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

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

Simplify each term.

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

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

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

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Write $\cos (x)$ as a fraction with denominator $1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Apply the distributive property.

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

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

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Write $\sin (x)$ as a fraction with denominator $1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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Factor out the greatest common factor $\sin (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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Factor out the greatest common factor $\cos (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

Simplify each term.

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

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

Separate fractions.

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

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

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

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

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

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