Trigonometry Examples

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

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

Step 1

Start on the left side.

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

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

Step 2

Apply the distributive property.

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

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

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)

$1 + \cos (A)$ by

1

$1$ .

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

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

Step 3.1.2

Apply the distributive property.

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

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

Step 3.1.3

Multiply

- \cos (A)

$- \cos (A)$ by

1

$1$ .

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

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

Step 3.1.4

Multiply

\cos (A) (- \cos (A))

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

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Step 3.1.4.1

Raise

\cos (A)

$\cos (A)$ to the power of

1

$1$ .

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

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

Step 3.1.4.2

Raise

\cos (A)

$\cos (A)$ to the power of

1

$1$ .

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

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

Step 3.1.4.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.1.4.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

Step 3.2

Subtract

\cos (A)

$\cos (A)$ from

\cos (A)

$\cos (A)$ .

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

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

Step 3.3

Add

1

$1$ and

0

$0$ .

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

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

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

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

Step 4

Apply pythagorean identity.

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

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

Step 5

Cancel the common factor of

{\sin (A)}^{2}

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

\sin (A)

$\sin (A)$ .

\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)

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