Trigonometry Examples

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

Step 1

Start on the left side.

$\cos^{4} (t) - \sin^{4} (t)$

Step 2

Factor.

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Step 2.1

Rewrite $\cos^{4} (t)$ as ${(\cos^{2} (t))}^{2}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Rearrange terms.

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

Step 2.4.2

Apply pythagorean identity.

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 3

Apply the distributive property.

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Multiply $\cos (t) \cos (t)$ .

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Add $1$ and $1$ .

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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Step 4.1.4.1

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

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

Step 4.1.4.2

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

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

Step 4.1.4.3

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

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

Step 4.1.4.4

Add $1$ and $1$ .

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

Step 4.2

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

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Step 4.2.1

Reorder $\cos (t)$ and $- 1$ .

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

Step 4.2.2

Subtract $\cos (t) \sin (t)$ from $\cos (t) \sin (t)$ .

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

Step 4.3

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

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

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

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

Step 7

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

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