Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify.

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Step 2.1

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

$(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$

Step 2.2

Expand $(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\sin (x) (\sin (x) + \cos (x)) + \cos (x) (\sin (x) + \cos (x))$

Step 2.2.2

Apply the distributive property.

$\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) (\sin (x) + \cos (x))$

Step 2.2.3

Apply the distributive property.

$\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x)$

Step 2.3

Simplify and combine like terms.

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

Step 3

Apply Pythagorean identity.

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Step 3.1

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

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

Step 3.2

Apply pythagorean identity.

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

Step 4

Rewrite $1 + 2 \cos (x) \sin (x)$ as $1 + 2 \sin (x) \cos (x)$ .

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

Step 5

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

${(\sin (x) + \cos (x))}^{2} = 1 + 2 \sin (x) \cos (x)$ is an identity