Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(1 + \sin (t))}^{2}$ as $(1 + \sin (t)) (1 + \sin (t))$ .

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

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

Step 1.3.1.2

Multiply $\sin (t)$ by $1$ .

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

Step 1.3.1.3

Multiply $\sin (t)$ by $1$ .

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

Step 1.3.1.4

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

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Step 1.3.1.4.1

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

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

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

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

Step 1.3.1.4.4

Add $1$ and $1$ .

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

Step 1.3.2

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

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

Step 2

Apply pythagorean identity.

$1 + 2 \sin (t) + 1$

Step 3

Add $1$ and $1$ .

$2 + 2 \sin (t)$