Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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Step 2.1

Multiply by $1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

Apply the distributive property.

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

Step 4.1.1.2

Apply the distributive property.

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

Step 4.1.1.3

Apply the distributive property.

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

Step 4.1.2

Simplify and combine like terms.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply $1$ by $1$ .

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

Step 4.1.2.1.2

Multiply $- 2 {\sin (x)}^{2}$ by $1$ .

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

Step 4.1.2.1.3

Multiply $- 1$ by $1$ .

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

Step 4.1.2.1.4

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

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Step 4.1.2.1.4.1

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

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

Step 4.1.2.1.4.2

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

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

Step 4.1.2.1.4.3

Add $2$ and $2$ .

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

Step 4.1.2.1.5

Multiply $- 2$ by $- 1$ .

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

Step 4.1.2.2

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

$1 - 3 {\sin (x)}^{2} + 2 {\sin (x)}^{4} - {\sin (x)}^{2} + 2 {\sin (x)}^{4}$

Step 4.2

Subtract ${\sin (x)}^{2}$ from $- 3 {\sin (x)}^{2}$ .

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

Step 4.3

Add $2 {\sin (x)}^{4}$ and $2 {\sin (x)}^{4}$ .

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

Step 5

Factor using the perfect square rule.

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Step 5.1

Rearrange terms.

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

Step 5.2

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

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

Step 5.3

Rewrite $4 \sin^{4} (x)$ as ${(2 \sin^{2} (x))}^{2}$ .

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

Step 5.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 \sin^{2} (x) = 2 \cdot 1 \cdot (2 \sin^{2} (x))$

Step 5.5

Rewrite the polynomial.

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

Step 5.6

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = 2 \sin^{2} (x)$ .

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

Step 6

Apply the cosine double-angle identity.

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

Step 7

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

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