The given mathematical expression is:

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

Let’s simplify it step by step.

Step 1: Recall the trigonometric identities

1. $\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}$
2. Use the sum to simplify the entire expression.

Step 2: Rewrite $\tan(\alpha)$ as $\frac{\sin(\alpha)}{\cos(\alpha)}$

The expression becomes: $\frac{\sin(\alpha)}{\cos(\alpha)} + \frac{\sin(\alpha)}{1 - \cos(\alpha)}$

Step 3: Combine the two terms under a common denominator

The common denominator is $\cos(\alpha)(1 - \cos(\alpha))$. So, rewrite each fraction:

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

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

Adding these, we get:

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

Step 4: Simplify the numerator

Expand and combine like terms in the numerator: $\sin(\alpha)(1 - \cos(\alpha)) + \sin(\alpha)\cos(\alpha) = \sin(\alpha) - \sin(\alpha)\cos(\alpha) + \sin(\alpha)\cos(\alpha)$

Notice that $-\sin(\alpha)\cos(\alpha) + \sin(\alpha)\cos(\alpha)$ cancels out. This leaves:

$\sin(\alpha)$

So the expression becomes:

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

Final Simplified Expression:

The simplified expression is:

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

Let me know if you would like further clarification or help!

Related Questions:

1. What is the significance of trigonometric identities in simplifying expressions?
2. How does combining fractions work when using common denominators in trigonometry?
3. Can this expression be evaluated for specific angles like $\alpha = 45^\circ$ or $\alpha = 30^\circ$?
4. What happens when $\cos(\alpha) = 1$ in this expression? Is it undefined?
5. How would the solution differ if $\tan(\alpha)$ had a coefficient, e.g., $2\tan(\alpha)$?

Tip:

Always look for opportunities to apply fundamental identities, like $\tan(x) = \frac{\sin(x)}{\cos(x)}$, to simplify trigonometric expressions!