Trigonometry Examples

${(5 \sin (x) + 5 \cos (x))}^{2} = 25 + 25 \sin (2 x)$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

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

$(5 \sin (x) + 5 \cos (x)) (5 \sin (x) + 5 \cos (x))$

Step 2.2

Expand $(5 \sin (x) + 5 \cos (x)) (5 \sin (x) + 5 \cos (x))$ using the FOIL Method.

Tap for more steps...

Step 2.2.1

Apply the distributive property.

$5 \sin (x) (5 \sin (x) + 5 \cos (x)) + 5 \cos (x) (5 \sin (x) + 5 \cos (x))$

Step 2.2.2

Apply the distributive property.

$5 \sin (x) (5 \sin (x)) + 5 \sin (x) (5 \cos (x)) + 5 \cos (x) (5 \sin (x) + 5 \cos (x))$

Step 2.2.3

Apply the distributive property.

$5 \sin (x) (5 \sin (x)) + 5 \sin (x) (5 \cos (x)) + 5 \cos (x) (5 \sin (x)) + 5 \cos (x) (5 \cos (x))$

Step 2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Multiply $5 \sin (x) (5 \sin (x))$ .

Tap for more steps...

Step 2.3.1.1.1

Multiply $5$ by $5$ .

$25 \sin (x) \sin (x) + 5 \sin (x) (5 \cos (x)) + 5 \cos (x) (5 \sin (x)) + 5 \cos (x) (5 \cos (x))$

Step 2.3.1.1.2

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

$25 (\sin^{1} (x) \sin (x)) + 5 \sin (x) (5 \cos (x)) + 5 \cos (x) (5 \sin (x)) + 5 \cos (x) (5 \cos (x))$

Step 2.3.1.1.3

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

$25 (\sin^{1} (x) \sin^{1} (x)) + 5 \sin (x) (5 \cos (x)) + 5 \cos (x) (5 \sin (x)) + 5 \cos (x) (5 \cos (x))$

Step 2.3.1.1.4

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

$25 {\sin (x)}^{1 + 1} + 5 \sin (x) (5 \cos (x)) + 5 \cos (x) (5 \sin (x)) + 5 \cos (x) (5 \cos (x))$

Step 2.3.1.1.5

Add $1$ and $1$ .

$25 \sin^{2} (x) + 5 \sin (x) (5 \cos (x)) + 5 \cos (x) (5 \sin (x)) + 5 \cos (x) (5 \cos (x))$

Step 2.3.1.2

Multiply $5$ by $5$ .

$25 \sin^{2} (x) + 25 \sin (x) \cos (x) + 5 \cos (x) (5 \sin (x)) + 5 \cos (x) (5 \cos (x))$

Step 2.3.1.3

Multiply $5$ by $5$ .

$25 \sin^{2} (x) + 25 \sin (x) \cos (x) + 25 \cos (x) \sin (x) + 5 \cos (x) (5 \cos (x))$

Step 2.3.1.4

Multiply $5 \cos (x) (5 \cos (x))$ .

Tap for more steps...

Step 2.3.1.4.1

Multiply $5$ by $5$ .

$25 \sin^{2} (x) + 25 \sin (x) \cos (x) + 25 \cos (x) \sin (x) + 25 \cos (x) \cos (x)$

Step 2.3.1.4.2

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

$25 \sin^{2} (x) + 25 \sin (x) \cos (x) + 25 \cos (x) \sin (x) + 25 (\cos^{1} (x) \cos (x))$

Step 2.3.1.4.3

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

$25 \sin^{2} (x) + 25 \sin (x) \cos (x) + 25 \cos (x) \sin (x) + 25 (\cos^{1} (x) \cos^{1} (x))$

Step 2.3.1.4.4

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

$25 \sin^{2} (x) + 25 \sin (x) \cos (x) + 25 \cos (x) \sin (x) + 25 {\cos (x)}^{1 + 1}$

Step 2.3.1.4.5

Add $1$ and $1$ .

$25 \sin^{2} (x) + 25 \sin (x) \cos (x) + 25 \cos (x) \sin (x) + 25 \cos^{2} (x)$

Step 2.3.2

Reorder the factors of $25 \sin (x) \cos (x)$ .

$25 \sin^{2} (x) + 25 \cos (x) \sin (x) + 25 \cos (x) \sin (x) + 25 \cos^{2} (x)$

Step 2.3.3

Add $25 \cos (x) \sin (x)$ and $25 \cos (x) \sin (x)$ .

$25 \sin^{2} (x) + 50 \cos (x) \sin (x) + 25 \cos^{2} (x)$

Step 2.4

Move $25 \cos^{2} (x)$ .

$25 \sin^{2} (x) + 25 \cos^{2} (x) + 50 \cos (x) \sin (x)$

Step 2.5

Factor $25$ out of $25 \sin^{2} (x)$ .

$25 (\sin^{2} (x)) + 25 \cos^{2} (x) + 50 \cos (x) \sin (x)$

Step 2.6

Factor $25$ out of $25 \cos^{2} (x)$ .

$25 (\sin^{2} (x)) + 25 (\cos^{2} (x)) + 50 \cos (x) \sin (x)$

Step 2.7

Factor $25$ out of $25 (\sin^{2} (x)) + 25 (\cos^{2} (x))$ .

$25 (\sin^{2} (x) + \cos^{2} (x)) + 50 \cos (x) \sin (x)$

Step 2.8

Apply pythagorean identity.

$25 \cdot 1 + 50 \cos (x) \sin (x)$

Step 2.9

Multiply $25$ by $1$ .

$25 + 50 \cos (x) \sin (x)$

Step 3

Factor.

Tap for more steps...

Step 3.1

Factor $25$ out of $25 + 50 \cos (x) \sin (x)$ .

Tap for more steps...

Step 3.1.1

Factor $25$ out of $25$ .

$25 (1) + 50 \cos (x) \sin (x)$

Step 3.1.2

Factor $25$ out of $50 \cos (x) \sin (x)$ .

$25 (1) + 25 (2 \cos (x) \sin (x))$

Step 3.1.3

Factor $25$ out of $25 (1) + 25 (2 \cos (x) \sin (x))$ .

$25 (1 + 2 \cos (x) \sin (x))$

Step 3.2

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

$25 (1 + \sin (x) (2 \cos (x)))$

Step 3.3

Remove parentheses.

$25 (1 + \sin (x) \cdot 2 \cos (x))$

Step 3.4

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

$25 (1 + 2 \cdot \sin (x) \cos (x))$

Step 3.5

Apply the sine double-angle identity.

$25 (1 + \sin (2 x))$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$25 \cdot 1 + 25 \sin (2 x)$

Step 4.2

Multiply $25$ by $1$ .

$25 + 25 \sin (2 x)$

Step 5

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

${(5 \sin (x) + 5 \cos (x))}^{2} = 25 + 25 \sin (2 x)$ is an identity