Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$ .

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Step 3.1.1

Reorder the factors in the terms $\cos (x) (- \sin (x))$ and $\sin (x) \cos (x)$ .

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

Step 3.1.2

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

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

Step 3.1.3

Add $\cos (x) \cos (x)$ and $0$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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

Step 3.2.1.4

Add $1$ and $1$ .

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.3.4

Add $1$ and $1$ .

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

Step 4

Apply the cosine double-angle identity.

$\cos (2 x)$

Step 5

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 6

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 7

Substitute the actual values of $a = \cos (2 x)$ and $b = 0$ .

$| z | = \sqrt{0^{2} + \cos^{2} (2 x)}$

Step 8

Find $| z |$ .

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Step 8.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + \cos^{2} (2 x)}$

Step 8.2

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

$| z | = \sqrt{\cos^{2} (2 x)}$

Step 8.3

Pull terms out from under the radical, assuming positive real numbers.

$| z | = \cos (2 x)$

Step 9

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{0}{\cos (2 x)})$

Step 10

Substitute the values of $θ = \arctan (\frac{0}{\cos (2 x)})$ and $| z | = \cos (2 x)$ .

$\cos (2 x) (\cos (\arctan (\frac{0}{\cos (2 x)})) + i \sin (\arctan (\frac{0}{\cos (2 x)})))$