Trigonometry Examples

$1 + (\sin (x) - \cos (x)) (\sin (x) + \cos (x))$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Expand $(\sin (x) - \cos (x)) (\sin (x) + \cos (x))$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$1 + \sin (x) (\sin (x) + \cos (x)) - \cos (x) (\sin (x) + \cos (x))$

Step 1.1.1.2

Apply the distributive property.

$1 + \sin (x) \sin (x) + \sin (x) \cos (x) - \cos (x) (\sin (x) + \cos (x))$

Step 1.1.1.3

Apply the distributive property.

$1 + \sin (x) \sin (x) + \sin (x) \cos (x) - \cos (x) \sin (x) - \cos (x) \cos (x)$

Step 1.1.2

Combine the opposite terms in $\sin (x) \sin (x) + \sin (x) \cos (x) - \cos (x) \sin (x) - \cos (x) \cos (x)$ .

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Step 1.1.2.1

Reorder the factors in the terms $\sin (x) \cos (x)$ and $- \cos (x) \sin (x)$ .

$1 + \sin (x) \sin (x) + \cos (x) \sin (x) - \cos (x) \sin (x) - \cos (x) \cos (x)$

Step 1.1.2.2

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

$1 + \sin (x) \sin (x) + 0 - \cos (x) \cos (x)$

Step 1.1.2.3

Add $\sin (x) \sin (x)$ and $0$ .

$1 + \sin (x) \sin (x) - \cos (x) \cos (x)$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

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

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

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

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

Step 1.1.3.1.4

Add $1$ and $1$ .

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

Step 1.1.3.2

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

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Step 1.1.3.2.1

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

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

Step 1.1.3.2.2

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

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

Step 1.1.3.2.3

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

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

Step 1.1.3.2.4

Add $1$ and $1$ .

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

Step 1.2

Move $- \cos^{2} (x)$ .

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

Step 2

Apply pythagorean identity.

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

Step 3

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

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