Trigonometry Examples

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t) = \sin (t) + \cos (t)$

Step 1

Start on the left side.

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

Step 2

Factor.

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Step 2.1

Since both terms are perfect cubes, factor using the sum of cubes formula,

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where

$a = \sin (t)$ and

$b = \cos (t)$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Rearrange terms.

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

Step 2.2.2

Apply pythagorean identity.

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Expand

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1)$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

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

Step 3.1.1.2

Apply the distributive property.

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

Step 3.1.1.3

Apply the distributive property.

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

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Multiply

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

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Step 3.1.2.1.1

Raise

$\sin (t)$ to the power of

$1$ .

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

Step 3.1.2.1.2

Raise

$\sin (t)$ to the power of

$1$ .

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

Step 3.1.2.1.3

Use the power rule

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

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

Step 3.1.2.1.4

Add

$1$ and

$1$ .

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

Step 3.1.2.2

Multiply

$\sin (t)$ by

$1$ .

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

Step 3.1.2.3

Multiply

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

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Step 3.1.2.3.1

Raise

$\cos (t)$ to the power of

$1$ .

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

Step 3.1.2.3.2

Raise

$\cos (t)$ to the power of

$1$ .

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

Step 3.1.2.3.3

Use the power rule

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

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

Step 3.1.2.3.4

Add

$1$ and

$1$ .

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

Step 3.1.2.4

Multiply

$\cos (t)$ by

$1$ .

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

Step 3.2

Add

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

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

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

Step 3.3

Add

$\sin (t)$ and

$0$ .

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

Step 3.4

Add

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

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

$0 + \cos (t) + \sin (t)$

Step 3.5

Add

$0$ and

$\cos (t)$ .

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

Step 4

Rewrite

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

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

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

Step 5

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

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t) = \sin (t) + \cos (t)$ is an identity