Trigonometry Examples

${(\sin (a) - \cos (a))}^{2} = 1 - 2 \sin (a) \cos (a)$

Step 1

Simplify ${(\sin (a) - \cos (a))}^{2}$ .

Tap for more steps...

Step 1.1

Rewrite ${(\sin (a) - \cos (a))}^{2}$ as $(\sin (a) - \cos (a)) (\sin (a) - \cos (a))$ .

$(\sin (a) - \cos (a)) (\sin (a) - \cos (a)) = 1 - 2 \sin (a) \cos (a)$

Step 1.2

Expand $(\sin (a) - \cos (a)) (\sin (a) - \cos (a))$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\sin (a) (\sin (a) - \cos (a)) - \cos (a) (\sin (a) - \cos (a)) = 1 - 2 \sin (a) \cos (a)$

Step 1.2.2

Apply the distributive property.

$\sin (a) \sin (a) + \sin (a) (- \cos (a)) - \cos (a) (\sin (a) - \cos (a)) = 1 - 2 \sin (a) \cos (a)$

Step 1.2.3

Apply the distributive property.

$\sin (a) \sin (a) + \sin (a) (- \cos (a)) - \cos (a) \sin (a) - \cos (a) (- \cos (a)) = 1 - 2 \sin (a) \cos (a)$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $\sin (a) \sin (a)$ .

Tap for more steps...

Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

Add $1$ and $1$ .

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

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$\sin^{2} (a) - \sin (a) \cos (a) - \cos (a) \sin (a) - \cos (a) (- \cos (a)) = 1 - 2 \sin (a) \cos (a)$

Step 1.3.1.3

Multiply $- \cos (a) (- \cos (a))$ .

Tap for more steps...

Step 1.3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.3.2

Multiply $\cos (a)$ by $1$ .

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

Step 1.3.1.3.3

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

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

Step 1.3.1.3.4

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

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

Step 1.3.1.3.5

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

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

Step 1.3.1.3.6

Add $1$ and $1$ .

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

Step 1.3.2

Reorder the factors of $- \sin (a) \cos (a)$ .

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

Step 1.3.3

Subtract $\cos (a) \sin (a)$ from $- \cos (a) \sin (a)$ .

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

Step 1.4

Move $\cos^{2} (a)$ .

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

Step 1.5

Apply pythagorean identity.

$1 - 2 \cos (a) \sin (a) = 1 - 2 \sin (a) \cos (a)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$a \approx 0, - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, - \frac{2}{5}$

Step 3