Trigonometry Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Convert from

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

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

\tan (x)

$\tan (x)$ .

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

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

Step 1.2

Convert from

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

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

\cot (x)

$\cot (x)$ .

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

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

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

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

Step 2

Apply the distributive property.

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

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

Step 3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.1

Reorder

\cos (x)

$\cos (x)$ and

\tan (x)

$\tan (x)$ .

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

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

Step 3.2

Rewrite

\cos (x) \tan (x)

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

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

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

Step 3.3

Cancel the common factors.

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

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

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

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