Trigonometry Examples

${(\cos (2 x) + \sin (2 x))}^{2} = 1 + \sin (4 x)$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

Expand $(\cos (2 x) + \sin (2 x)) (\cos (2 x) + \sin (2 x))$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $\cos (2 x) \cos (2 x)$ .

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Step 2.3.1.1.1

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 2.3.1.1.2

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 2.3.1.1.3

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

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

Step 2.3.1.1.4

Add $1$ and $1$ .

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

Step 2.3.1.2

Multiply $\sin (2 x) \sin (2 x)$ .

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Step 2.3.1.2.1

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 2.3.1.2.2

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 2.3.1.2.3

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

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

Step 2.3.1.2.4

Add $1$ and $1$ .

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

Step 2.3.2

Reorder the factors of $\sin (2 x) \cos (2 x)$ .

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

Step 2.3.3

Add $\cos (2 x) \sin (2 x)$ and $\cos (2 x) \sin (2 x)$ .

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

Step 2.4

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

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

Step 2.5

Rearrange terms.

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

Step 2.6

Apply pythagorean identity.

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

Step 2.7

Simplify each term.

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Step 2.7.1

Reorder $2 \cos (2 x)$ and $\sin (2 x)$ .

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

Step 2.7.2

Reorder $\sin (2 x)$ and $2$ .

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

Step 2.7.3

Apply the sine double-angle identity.

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

Step 2.7.4

Multiply $2$ by $2$ .

$1 + \sin (4 x)$

Step 3

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

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