Calculus Examples

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

Step 1

To write $\frac{1}{1 - \sin (t)}$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (t)}{1 + \sin (t)}$ .

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

Step 2

To write $\frac{1}{1 + \sin (t)}$ as a fraction with a common denominator, multiply by $\frac{1 - \sin (t)}{1 - \sin (t)}$ .

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

Step 3

Write each expression with a common denominator of $(1 - \sin (t)) (1 + \sin (t))$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(1 - \sin (t)) (1 + \sin (t))$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Add $1$ and $1$ .

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

Step 5.2

Subtract $\sin (t)$ from $\sin (t)$ .

$\frac{2 + 0}{(1 + \sin (t)) (1 - \sin (t))}$

Step 5.3

Add $2$ and $0$ .

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