Trigonometry Examples

$\sin^{3} (x) \cos^{4} (x) = (\cos^{4} (x) - \cos^{6} (x)) \sin (x)$

Step 1

Start on the right side.

$(\cos^{4} (x) - \cos^{6} (x)) \sin (x)$

Step 2

Simplify the expression.

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Step 2.1

Apply the distributive property.

$\cos^{4} (x) \sin (x) - \cos^{6} (x) \sin (x)$

Step 2.2

Factor $\cos^{4} (x) \sin (x)$ out of $\cos^{4} (x) \sin (x) - \cos^{6} (x) \sin (x)$ .

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Step 2.2.1

Factor $\cos^{4} (x) \sin (x)$ out of $\cos^{4} (x) \sin (x)$ .

$\cos^{4} (x) \sin (x) (1) - \cos^{6} (x) \sin (x)$

Step 2.2.2

Factor $\cos^{4} (x) \sin (x)$ out of $- \cos^{6} (x) \sin (x)$ .

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

Step 2.2.3

Factor $\cos^{4} (x) \sin (x)$ out of $\cos^{4} (x) \sin (x) (1) + \cos^{4} (x) \sin (x) (- \cos^{2} (x))$ .

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

Step 2.3

Apply pythagorean identity.

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

Step 2.4

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 2.4.1

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

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

Step 2.4.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.3

Add $2$ and $1$ .

$\cos^{4} (x) \sin^{3} (x)$

Step 3

Rewrite $\cos^{4} (x) \sin^{3} (x)$ as $\sin^{3} (x) \cos^{4} (x)$ .

$\sin^{3} (x) \cos^{4} (x)$

Step 4

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

$\sin^{3} (x) \cos^{4} (x) = (\cos^{4} (x) - \cos^{6} (x)) \sin (x)$ is an identity