Precalculus Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(\sin (x))}^{2 - k} \cdot {(\cos (x))}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(\sin (x))}^{2 - 0} \cdot \cos^{0} (x) + \frac{2!}{(2 - 1)! 1!} \cdot {(\sin (x))}^{2 - 1} \cdot \cos^{1} (x) + \frac{2!}{(2 - 2)! 2!} \cdot {(\sin (x))}^{2 - 2} \cdot \cos^{2} (x)$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot \sin^{2} (x) \cdot \cos^{0} (x) + 2 \cdot \sin^{1} (x) \cdot \cos^{1} (x) + 1 \cdot \sin^{0} (x) \cdot \cos^{2} (x)$

Step 4

Simplify the polynomial result.

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Step 4.1

Simplify with factoring out.

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Step 4.1.1

Factor $\sin^{0} (x) \cos^{0} (x)$ out of $1 \cdot \sin^{2} (x) \cdot \cos^{0} (x) + 2 \cdot \sin^{1} (x) \cdot \cos^{1} (x) + 1 \cdot \sin^{0} (x) \cdot \cos^{2} (x)$ .

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Step 4.1.1.1

Factor $\sin^{0} (x) \cos^{0} (x)$ out of $1 \cdot \sin^{2} (x) \cdot \cos^{0} (x)$ .

$\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x)) + 2 \cdot \sin^{1} (x) \cdot \cos^{1} (x) + 1 \cdot \sin^{0} (x) \cdot \cos^{2} (x)$

Step 4.1.1.2

Factor $\sin^{0} (x) \cos^{0} (x)$ out of $2 \cdot \sin^{1} (x) \cdot \cos^{1} (x)$ .

$\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x)) + \sin^{0} (x) \cos^{0} (x) (2 \cdot \sin (x) \cdot \cos (x)) + 1 \cdot \sin^{0} (x) \cdot \cos^{2} (x)$

Step 4.1.1.3

Factor $\sin^{0} (x) \cos^{0} (x)$ out of $1 \cdot \sin^{0} (x) \cdot \cos^{2} (x)$ .

$\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x)) + \sin^{0} (x) \cos^{0} (x) (2 \cdot \sin (x) \cdot \cos (x)) + \sin^{0} (x) \cos^{0} (x) (1 \cdot \cos^{2} (x))$

Step 4.1.1.4

Factor $\sin^{0} (x) \cos^{0} (x)$ out of $\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x)) + \sin^{0} (x) \cos^{0} (x) (2 \cdot \sin (x) \cdot \cos (x))$ .

$\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x) + 2 \cdot \sin (x) \cdot \cos (x)) + \sin^{0} (x) \cos^{0} (x) (1 \cdot \cos^{2} (x))$

Step 4.1.1.5

Factor $\sin^{0} (x) \cos^{0} (x)$ out of $\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x) + 2 \cdot \sin (x) \cdot \cos (x)) + \sin^{0} (x) \cos^{0} (x) (1 \cdot \cos^{2} (x))$ .

$\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x) + 2 \cdot \sin (x) \cdot \cos (x) + 1 \cdot \cos^{2} (x))$

Step 4.1.2

Move $1 \cdot \cos^{2} (x)$ .

$\sin^{0} (x) \cos^{0} (x) (1 \cdot \sin^{2} (x) + 1 \cdot \cos^{2} (x) + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.1.3

Factor $1$ out of $1 \cdot \sin^{2} (x)$ .

$\sin^{0} (x) \cos^{0} (x) (1 (\sin^{2} (x)) + 1 \cdot \cos^{2} (x) + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.1.4

Factor $1$ out of $1 \cdot \cos^{2} (x)$ .

$\sin^{0} (x) \cos^{0} (x) (1 (\sin^{2} (x)) + 1 (\cos^{2} (x)) + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.1.5

Factor $1$ out of $1 (\sin^{2} (x)) + 1 (\cos^{2} (x))$ .

$\sin^{0} (x) \cos^{0} (x) (1 (\sin^{2} (x) + \cos^{2} (x)) + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.2

Apply pythagorean identity.

$\sin^{0} (x) \cos^{0} (x) (1 \cdot 1 + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.3

Simplify terms.

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Step 4.3.1

Anything raised to $0$ is $1$ .

$1 \cos^{0} (x) (1 \cdot 1 + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Multiply $\cos^{0} (x)$ by $1$ .

$\cos^{0} (x) (1 \cdot 1 + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.3.2.2

Anything raised to $0$ is $1$ .

$1 (1 \cdot 1 + 2 \cdot \sin (x) \cdot \cos (x))$

Step 4.3.2.3

Multiply $1 \cdot 1 + 2 \cdot \sin (x) \cdot \cos (x)$ by $1$ .

$1 \cdot 1 + 2 \cdot \sin (x) \cdot \cos (x)$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Multiply $1$ by $1$ .

$1 + 2 \cdot \sin (x) \cdot \cos (x)$

Step 4.3.3.2

Apply the sine double-angle identity.

$1 + \sin (2 x)$