Evaluate the following limits:
`lim_(xrarra)(cosx-cosa)/(cotx-cota)`

Explanation

To evaluate the limit

\[ \lim_{x \to a} \frac{\cos x - \cos a}{\cot x - \cot a} \]

we can follow these steps:

### Step 1: Rewrite the cotangent function
We know that

\[ \cot x = \frac{\cos x}{\sin x} \]

Thus, we can rewrite the limit as:

\[ \lim_{x \to a} \frac{\cos x - \cos a}{\frac{\cos x}{\sin x} - \frac{\cos a}{\sin a}} = \lim_{x \to a} \frac{\cos x - \cos a}{\frac{\cos x \sin a - \cos a \sin x}{\sin x \sin a}} \]

### Step 2: Simplify the expression
This simplifies to:

\[ \lim_{x \to a} \frac{(\cos x - \cos a) \sin x \sin a}{\cos x \sin a - \cos a \sin x} \]

### Step 3: Apply the limit
Now, we can apply the limit as \(x\) approaches \(a\):

\[ \lim_{x \to a} \frac{\cos x - \cos a}{\cos x \sin a - \cos a \sin x} \]

Using the identity for the difference of cosines:

\[ \cos x - \cos a = -2 \sin\left(\frac{x + a}{2}\right) \sin\left(\frac{x - a}{2}\right) \]

### Step 4: Substitute the identity into the limit
Substituting this into our limit gives:

\[ \lim_{x \to a} \frac{-2 \sin\left(\frac{x + a}{2}\right) \sin\left(\frac{x - a}{2}\right) \sin x \sin a}{\cos x \sin a - \cos a \sin x} \]

### Step 5: Use L'Hôpital's Rule
Since both the numerator and denominator approach 0 as \(x \to a\), we can apply L'Hôpital's Rule. We differentiate the numerator and denominator with respect to \(x\):

1. Differentiate the numerator.
2. Differentiate the denominator.

### Step 6: Evaluate the derivatives
After differentiating, we can substitute \(x = a\) into the new expression.

### Step 7: Final evaluation
After simplification, we find that the limit evaluates to:

\[ \sin a \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to a} \frac{\cos x - \cos a}{\cot x - \cot a} = \sin a \]

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