Trigonometry Examples

$\frac{(1 + \cos (A)) (1 - \cos (A))}{\sin (A)} = \sin (A)$

Step 1

Start on the left side.

$\frac{(1 + \cos (A)) (1 - \cos (A))}{\sin (A)}$

Step 2

Apply the distributive property.

$\frac{(1 + \cos (A)) \cdot 1 + (1 + \cos (A)) (- \cos (A))}{\sin (A)}$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1 + \cos (A)$ by $1$ .

$\frac{1 + \cos (A) + (1 + \cos (A)) (- \cos (A))}{\sin (A)}$

Step 3.1.2

Apply the distributive property.

$\frac{1 + \cos (A) + 1 (- \cos (A)) + \cos (A) (- \cos (A))}{\sin (A)}$

Step 3.1.3

Multiply $- \cos (A)$ by $1$ .

$\frac{1 + \cos (A) - \cos (A) + \cos (A) (- \cos (A))}{\sin (A)}$

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

Add $1$ and $1$ .

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

Step 3.2

Subtract $\cos (A)$ from $\cos (A)$ .

$\frac{1 + 0 - {\cos (A)}^{2}}{\sin (A)}$

Step 3.3

Add $1$ and $0$ .

$\frac{1 - \cos^{2} (A)}{\sin (A)}$

Step 4

Apply pythagorean identity.

$\frac{\sin^{2} (A)}{\sin (A)}$

Step 5

Cancel the common factor of ${\sin (A)}^{2}$ and $\sin (A)$ .

$\sin (A)$

Step 6

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

$\frac{(1 + \cos (A)) (1 - \cos (A))}{\sin (A)} = \sin (A)$ is an identity