Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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Step 2.2.1

Factor $\cos (x)$ out of $\sin (x) \cos (x)$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

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

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Step 7.1

Factor $\sin (x)$ out of $\sin (x) \cos (x)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Add $1$ and $1$ .

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

Step 12

Apply pythagorean identity.

$1$