Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 1.1.2

Since $\tan (- x)$ is an odd function, rewrite $\tan (- x)$ as $- \tan (x)$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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Step 1.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.4.2

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $\cos (x) \cos (x)$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Add $1$ and $1$ .

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

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 2

Combine the opposite terms in $\cos^{2} (x) + \sin (x) - \sin (x)$ .

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Step 2.1

Subtract $\sin (x)$ from $\sin (x)$ .

$\cos^{2} (x) + 0$

Step 2.2

Add $\cos^{2} (x)$ and $0$ .

$\cos^{2} (x)$