Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Expand $(1 - \cos (x)) (1 + \cos (x))$ using the FOIL Method.

Tap for more steps...

Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Apply the distributive property.

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

Step 2.1.3

Apply the distributive property.

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

Step 2.2

Simplify and combine like terms.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $1$ by $1$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.2.1.4

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

Tap for more steps...

Step 2.2.1.4.1

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

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

Step 2.2.1.4.2

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

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

Step 2.2.1.4.3

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

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

Step 2.2.1.4.4

Add $1$ and $1$ .

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

Step 2.2.2

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

$1 + 0 - \cos^{2} (x)$

Step 2.2.3

Add $1$ and $0$ .

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

Step 2.3

Apply pythagorean identity.

$\sin^{2} (x)$

Step 3

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

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