Evaluate the following limits :
`lim_(theta to 0)((cosectheta-cottheta)/(theta))`

Answer

To evaluate the limit

\[ \lim_{\theta \to 0} \frac{\csc \theta - \cot \theta}{\theta} \]

we will follow these steps:

### Step 1: Rewrite the functions in terms of sine and cosine
Recall that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Therefore, we can rewrite the expression as:

\[ \csc \theta - \cot \theta = \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} = \frac{1 - \cos \theta}{\sin \theta} \]

### Step 2: Substitute back into the limit
Now, substituting this back into our limit gives:

\[ \lim_{\theta \to 0} \frac{\frac{1 - \cos \theta}{\sin \theta}}{\theta} = \lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta \sin \theta} \]

### Step 3: Use the identity for \(1 - \cos \theta\)
We can use the trigonometric identity \(1 - \cos \theta = 2 \sin^2\left(\frac{\theta}{2}\right)\):

\[ \lim_{\theta \to 0} \frac{2 \sin^2\left(\frac{\theta}{2}\right)}{\theta \sin \theta} \]

### Step 4: Simplify the limit
Now, we can express \(\sin \theta\) in terms of \(\sin\left(\frac{\theta}{2}\right)\):

\[ \sin \theta = 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \]

Substituting this into our limit gives:

\[ \lim_{\theta \to 0} \frac{2 \sin^2\left(\frac{\theta}{2}\right)}{\theta \cdot 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)} = \lim_{\theta \to 0} \frac{\sin\left(\frac{\theta}{2}\right)}{\theta \cos\left(\frac{\theta}{2}\right)} \]

### Step 5: Change of variable
Let \(x = \frac{\theta}{2}\), then as \(\theta \to 0\), \(x \to 0\) and \(\theta = 2x\):

\[ \lim_{x \to 0} \frac{\sin x}{2x \cos x} = \frac{1}{2} \lim_{x \to 0} \frac{\sin x}{x} \cdot \frac{1}{\cos x} \]

### Step 6: Evaluate the limit
Using the limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\lim_{x \to 0} \cos x = 1\):

\[ = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2} \]

Thus, the final answer is:

\[ \lim_{\theta \to 0} \frac{\csc \theta - \cot \theta}{\theta} = \frac{1}{2} \]