Trigonometry Examples

$(1 - \cos^{2} (x)) (1 + \cos^{2} (x)) = 2 \sin^{2} (x) - \sin^{4} (x)$

Step 1

Start on the left side.

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

Step 2

Factor.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $1$ by $1$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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Step 4.2.1.4.1

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

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

Step 4.2.1.4.2

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

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

Step 4.2.1.4.3

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

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

Step 4.2.1.4.4

Add $1$ and $1$ .

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

Step 4.2.2

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

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

Step 4.2.3

Add $1$ and $0$ .

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

Step 4.3

Add $1$ and $1$ .

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

Step 4.4

Expand $(1 - {\cos (x)}^{2}) (2 - {\sin (x)}^{2})$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

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

Step 4.4.2

Apply the distributive property.

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

Step 4.4.3

Apply the distributive property.

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

Step 4.5

Simplify each term.

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

Factor $2$ out of $2$ .

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

Step 5.3

Factor $2$ out of $- 2 \cos^{2} (x)$ .

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

Step 5.4

Factor $2$ out of $2 (1) + 2 (- \cos^{2} (x))$ .

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

Step 5.5

Apply pythagorean identity.

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

Step 5.6

Factor $\sin^{2} (x)$ out of $2 \sin^{2} (x) - \sin^{2} (x) + \cos^{2} (x) \sin^{2} (x)$ .

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Step 5.6.1

Factor $\sin^{2} (x)$ out of $2 \sin^{2} (x)$ .

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

Step 5.6.2

Factor $\sin^{2} (x)$ out of $- \sin^{2} (x)$ .

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

Step 5.6.3

Factor $\sin^{2} (x)$ out of $\cos^{2} (x) \sin^{2} (x)$ .

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

Step 5.6.4

Factor $\sin^{2} (x)$ out of $\sin^{2} (x) \cdot 2 + \sin^{2} (x) \cdot - 1$ .

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

Step 5.6.5

Factor $\sin^{2} (x)$ out of $\sin^{2} (x) \cdot (2 - 1) + \sin^{2} (x) \cos^{2} (x)$ .

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

Step 5.7

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 5.8

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 5.9

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 5.10

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

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

Step 5.11

Apply pythagorean identity.

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

Step 5.12

Apply the distributive property.

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

Step 5.13

Move $2$ to the left of $\sin^{2} (x)$ .

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

Step 5.14

Rewrite using the commutative property of multiplication.

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

Step 5.15

Multiply $\sin^{2} (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 5.15.1

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

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

Step 5.15.2

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

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

Step 5.15.3

Add $2$ and $2$ .

$2 \sin^{2} (x) - \sin^{4} (x)$

Step 6

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

$(1 - \cos^{2} (x)) (1 + \cos^{2} (x)) = 2 \sin^{2} (x) - \sin^{4} (x)$ is an identity