Algebra Examples

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

Step 1

Rewrite $\csc (t)$ in terms of sines and cosines.

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

Step 2

Multiply the numerator and denominator of the fraction by $\sin (t)$ .

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $\sin (t)$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.4

Cancel the common factor of $\sin (t) + 1$ and $1 + \sin (t)$ .

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Step 4.4.1

Reorder terms.

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

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

Step 5

Convert from $\frac{1}{\sin (t)}$ to $\csc (t)$ .

$\csc (t)$