Trigonometry Examples

$\cos (6 y) = \cos^{2} (3 y) - \sin^{2} (3 y)$

Step 1

Simplify $\cos^{2} (3 y) - \sin^{2} (3 y)$ .

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Step 1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (3 y)$ and $b = \sin (3 y)$ .

$\cos (6 y) = (\cos (3 y) + \sin (3 y)) (\cos (3 y) - \sin (3 y))$

Step 1.2

Expand $(\cos (3 y) + \sin (3 y)) (\cos (3 y) - \sin (3 y))$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\cos (6 y) = \cos (3 y) (\cos (3 y) - \sin (3 y)) + \sin (3 y) (\cos (3 y) - \sin (3 y))$

Step 1.2.2

Apply the distributive property.

$\cos (6 y) = \cos (3 y) \cos (3 y) + \cos (3 y) (- \sin (3 y)) + \sin (3 y) (\cos (3 y) - \sin (3 y))$

Step 1.2.3

Apply the distributive property.

$\cos (6 y) = \cos (3 y) \cos (3 y) + \cos (3 y) (- \sin (3 y)) + \sin (3 y) \cos (3 y) + \sin (3 y) (- \sin (3 y))$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine the opposite terms in $\cos (3 y) \cos (3 y) + \cos (3 y) (- \sin (3 y)) + \sin (3 y) \cos (3 y) + \sin (3 y) (- \sin (3 y))$ .

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Step 1.3.1.1

Reorder the factors in the terms $\cos (3 y) (- \sin (3 y))$ and $\sin (3 y) \cos (3 y)$ .

$\cos (6 y) = \cos (3 y) \cos (3 y) - \cos (3 y) \sin (3 y) + \cos (3 y) \sin (3 y) + \sin (3 y) (- \sin (3 y))$

Step 1.3.1.2

Add $- \cos (3 y) \sin (3 y)$ and $\cos (3 y) \sin (3 y)$ .

$\cos (6 y) = \cos (3 y) \cos (3 y) + 0 + \sin (3 y) (- \sin (3 y))$

Step 1.3.1.3

Add $\cos (3 y) \cos (3 y)$ and $0$ .

$\cos (6 y) = \cos (3 y) \cos (3 y) + \sin (3 y) (- \sin (3 y))$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Multiply $\cos (3 y) \cos (3 y)$ .

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Step 1.3.2.1.1

Raise $\cos (3 y)$ to the power of $1$ .

$\cos (6 y) = \cos^{1} (3 y) \cos (3 y) + \sin (3 y) (- \sin (3 y))$

Step 1.3.2.1.2

Raise $\cos (3 y)$ to the power of $1$ .

$\cos (6 y) = \cos^{1} (3 y) \cos^{1} (3 y) + \sin (3 y) (- \sin (3 y))$

Step 1.3.2.1.3

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

$\cos (6 y) = {\cos (3 y)}^{1 + 1} + \sin (3 y) (- \sin (3 y))$

Step 1.3.2.1.4

Add $1$ and $1$ .

$\cos (6 y) = \cos^{2} (3 y) + \sin (3 y) (- \sin (3 y))$

Step 1.3.2.2

Rewrite using the commutative property of multiplication.

$\cos (6 y) = \cos^{2} (3 y) - \sin (3 y) \sin (3 y)$

Step 1.3.2.3

Multiply $- \sin (3 y) \sin (3 y)$ .

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Step 1.3.2.3.1

Raise $\sin (3 y)$ to the power of $1$ .

$\cos (6 y) = \cos^{2} (3 y) - (\sin^{1} (3 y) \sin (3 y))$

Step 1.3.2.3.2

Raise $\sin (3 y)$ to the power of $1$ .

$\cos (6 y) = \cos^{2} (3 y) - (\sin^{1} (3 y) \sin^{1} (3 y))$

Step 1.3.2.3.3

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

$\cos (6 y) = \cos^{2} (3 y) - {\sin (3 y)}^{1 + 1}$

Step 1.3.2.3.4

Add $1$ and $1$ .

$\cos (6 y) = \cos^{2} (3 y) - \sin^{2} (3 y)$

Step 1.4

Apply the cosine double-angle identity.

$\cos (6 y) = \cos (2 (3 y))$

Step 1.5

Multiply $3$ by $2$ .

$\cos (6 y) = \cos (6 y)$

Step 2

For the two functions to be equal, the arguments of each must be equal.

$6 y = 6 y$

Step 3

Move all terms containing $y$ to the left side of the equation.

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Step 3.1

Subtract $6 y$ from both sides of the equation.

$6 y - 6 y = 0$

Step 3.2

Subtract $6 y$ from $6 y$ .

$0 = 0$

Step 4

Since $0 = 0$ , the equation will always be true for any value of $y$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$