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

$1$ .

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

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

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

$1$ .

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

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

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

$1$ .

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

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

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

Step 2

Rearrange terms.

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

Step 3

Apply pythagorean identity.

$1$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$