Trigonometry Examples

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

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = cos (x)

$a = \cos (x)$ and

b = sin (x)

$b = \sin (x)$ .

(cos (x) + sin (x)) (cos (x) - sin (x))

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

Step 2

Expand

(cos (x) + sin (x)) (cos (x) - sin (x))

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

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Step 2.1

Apply the distributive property.

cos (x) (cos (x) - sin (x)) + sin (x) (cos (x) - sin (x))

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

Step 2.2

Apply the distributive property.

cos (x) cos (x) + cos (x) (- sin (x)) + sin (x) (cos (x) - sin (x))

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

Step 2.3

Apply the distributive property.

cos (x) cos (x) + cos (x) (- sin (x)) + sin (x) cos (x) + sin (x) (- sin (x))

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

cos (x) cos (x) + cos (x) (- sin (x)) + sin (x) cos (x) + sin (x) (- sin (x))

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

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in

cos (x) cos (x) + cos (x) (- sin (x)) + sin (x) cos (x) + sin (x) (- sin (x))

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

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Step 3.1.1

Reorder the factors in the terms

cos (x) (- sin (x))

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

sin (x) cos (x)

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

cos (x) cos (x) - cos (x) sin (x) + cos (x) sin (x) + sin (x) (- sin (x))

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

Step 3.1.2

Add

- cos (x) sin (x)

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

cos (x) sin (x)

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

cos (x) cos (x) + 0 + sin (x) (- sin (x))

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

Step 3.1.3

Add

cos (x) cos (x)

$\cos (x) \cos (x)$ and

0

$0$ .

cos (x) cos (x) + sin (x) (- sin (x))

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

cos (x) cos (x) + sin (x) (- sin (x))

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply

cos (x) cos (x)

$\cos (x) \cos (x)$ .

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Step 3.2.1.1

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 3.2.1.2

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 3.2.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.2.1.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

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

Step 3.2.3

Multiply

- sin (x) sin (x)

$- \sin (x) \sin (x)$ .

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Step 3.2.3.1

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 3.2.3.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 3.2.3.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.2.3.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

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

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

Step 4

Apply the cosine double-angle identity.

cos (2 x)

$\cos (2 x)$