Trigonometry Examples

$\sin^{4} (5 x) + \cos^{4} (5 x) + 2 \sin^{2} (5 x) \cos^{2} (5 x) = 1$

Step 1

Start on the left side.

$\sin^{4} (5 x) + \cos^{4} (5 x) + 2 \sin^{2} (5 x) \cos^{2} (5 x)$

Step 2

Factor.

Tap for more steps...

Step 2.1

Factor using the perfect square rule.

Tap for more steps...

Step 2.1.1

Rearrange terms.

$\sin^{4} (5 x) + 2 \sin^{2} (5 x) \cos^{2} (5 x) + \cos^{4} (5 x)$

Step 2.1.2

Rewrite $\sin^{4} (5 x)$ as ${(\sin^{2} (5 x))}^{2}$ .

${(\sin^{2} (5 x))}^{2} + 2 \sin^{2} (5 x) \cos^{2} (5 x) + \cos^{4} (5 x)$

Step 2.1.3

Rewrite $\cos^{4} (5 x)$ as ${(\cos^{2} (5 x))}^{2}$ .

${(\sin^{2} (5 x))}^{2} + 2 \sin^{2} (5 x) \cos^{2} (5 x) + {(\cos^{2} (5 x))}^{2}$

Step 2.1.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \sin^{2} (5 x) \cos^{2} (5 x) = 2 \cdot \sin^{2} (5 x) \cdot \cos^{2} (5 x)$

Step 2.1.5

Rewrite the polynomial.

${(\sin^{2} (5 x))}^{2} + 2 \cdot \sin^{2} (5 x) \cdot \cos^{2} (5 x) + {(\cos^{2} (5 x))}^{2}$

Step 2.1.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \sin^{2} (5 x)$ and $b = \cos^{2} (5 x)$ .

${(\sin^{2} (5 x) + \cos^{2} (5 x))}^{2}$

Step 2.2

Apply pythagorean identity.

$1^{2}$

Step 3

One to any power is one.

$1$

Step 4

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

$\sin^{4} (5 x) + \cos^{4} (5 x) + 2 \sin^{2} (5 x) \cos^{2} (5 x) = 1$ is an identity