Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Add $1$ and $1$ .

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

Step 1.3

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

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

Step 1.4

Multiply $- - \sin (x)$ .

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Step 1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.4.2

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

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

Step 1.5

Multiply $\sin (x) \sin (x)$ .

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

Add $1$ and $1$ .

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

Step 2

Rearrange terms.

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

Step 3

Apply pythagorean identity.

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