Trigonometry Examples

$(\sin (θ) - \cos (θ)) - {(\sin (θ) + \cos (θ))}^{2}$

Step 1

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

$\sin (θ) - \cos (θ) - ((\sin (θ) + \cos (θ)) (\sin (θ) + \cos (θ)))$

Step 2

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

Tap for more steps...

Step 2.1

Apply the distributive property.

$\sin (θ) - \cos (θ) - (\sin (θ) (\sin (θ) + \cos (θ)) + \cos (θ) (\sin (θ) + \cos (θ)))$

Step 2.2

Apply the distributive property.

$\sin (θ) - \cos (θ) - (\sin (θ) \sin (θ) + \sin (θ) \cos (θ) + \cos (θ) (\sin (θ) + \cos (θ)))$

Step 2.3

Apply the distributive property.

$\sin (θ) - \cos (θ) - (\sin (θ) \sin (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ))$

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $\sin (θ) \sin (θ)$ .

Tap for more steps...

Step 3.1.1.1

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

$\sin (θ) - \cos (θ) - (\sin^{1} (θ) \sin (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ))$

Step 3.1.1.2

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

$\sin (θ) - \cos (θ) - (\sin^{1} (θ) \sin^{1} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ))$

Step 3.1.1.3

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

$\sin (θ) - \cos (θ) - ({\sin (θ)}^{1 + 1} + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ))$

Step 3.1.1.4

Add $1$ and $1$ .

$\sin (θ) - \cos (θ) - (\sin^{2} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos (θ) \cos (θ))$

Step 3.1.2

Multiply $\cos (θ) \cos (θ)$ .

Tap for more steps...

Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Add $1$ and $1$ .

$\sin (θ) - \cos (θ) - (\sin^{2} (θ) + \sin (θ) \cos (θ) + \cos (θ) \sin (θ) + \cos^{2} (θ))$

Step 3.2

Reorder the factors of $\sin (θ) \cos (θ)$ .

$\sin (θ) - \cos (θ) - (\sin^{2} (θ) + \cos (θ) \sin (θ) + \cos (θ) \sin (θ) + \cos^{2} (θ))$

Step 3.3

Add $\cos (θ) \sin (θ)$ and $\cos (θ) \sin (θ)$ .

$\sin (θ) - \cos (θ) - (\sin^{2} (θ) + 2 \cos (θ) \sin (θ) + \cos^{2} (θ))$

Step 4

Move $\cos^{2} (θ)$ .

$\sin (θ) - \cos (θ) - (\sin^{2} (θ) + \cos^{2} (θ) + 2 \cos (θ) \sin (θ))$

Step 5

Apply pythagorean identity.

$\sin (θ) - \cos (θ) - (1 + 2 \cos (θ) \sin (θ))$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Reorder $2 \cos (θ)$ and $\sin (θ)$ .

$\sin (θ) - \cos (θ) - (1 + \sin (θ) (2 \cos (θ)))$

Step 6.2

Reorder $\sin (θ)$ and $2$ .

$\sin (θ) - \cos (θ) - (1 + 2 \cdot \sin (θ) \cos (θ))$

Step 6.3

Apply the sine double-angle identity.

$\sin (θ) - \cos (θ) - (1 + \sin (2 θ))$

Step 7

Apply the distributive property.

$\sin (θ) - \cos (θ) - 1 \cdot 1 - \sin (2 θ)$

Step 8

Multiply $- 1$ by $1$ .

$\sin (θ) - \cos (θ) - 1 - \sin (2 θ)$