Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Separate fractions.

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

Step 4.3

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

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

Step 4.4

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

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