Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Add $1$ and $1$ .

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

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

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

Tap for more steps...

Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

Add $1$ and $1$ .

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

Step 3.2

Add $- \cos (x) \sin (x)$ and $\cos (x) \sin (x)$ .

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

Step 3.3

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

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

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

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

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$(\sin (x) - \cos (x)) (\sin (x) + \cos (x)) = 1 - 2 \cos^{2} (x)$ is an identity