Precalculus Examples

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

Step 1

Start on the right side.

$\frac{\sin (t)}{1 + \cos (t)}$

Step 2

Multiply $\frac{\sin (t)}{1 + \cos (t)}$ by $\frac{1 - \cos (t)}{1 - \cos (t)}$ .

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\sin (t)$ by $1$ .

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

Step 4.3

Reorder factors in $\sin (t) + \sin (t) (- \cos (t))$ .

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

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 + \cos (t)) (1 - \cos (t))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply pythagorean identity.

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

Step 7

Simplify.

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Step 7.1

Factor $\sin (t)$ out of $\sin (t) - \sin (t) \cos (t)$ .

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Step 7.1.1

Multiply by $1$ .

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

Step 7.1.2

Factor $\sin (t)$ out of $- \sin (t) \cos (t)$ .

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

Step 7.1.3

Factor $\sin (t)$ out of $\sin (t) \cdot 1 + \sin (t) (- \cos (t))$ .

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

Step 7.2

Cancel the common factors.

$\frac{1 - \cos (t)}{\sin (t)}$

Step 8

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

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