`lim_(xrarrpi//2)(cos((pi)/(2)sin^(2)x))/(cot((pi)/(2)sin^(2)x))` equals

Options

A. `pi`
B. `-pi`
C. 1
D. `-1`

Answer

To solve the limit \( \lim_{x \to \frac{\pi}{2}} \frac{\cos\left(\frac{\pi}{2} \sin^2 x\right)}{\cot\left(\frac{\pi}{2} \sin^2 x\right)} \), we can follow these steps:

### Step 1: Rewrite the cotangent function
The cotangent function can be rewritten in terms of sine and cosine:
\[ \cot\left(\frac{\pi}{2} \sin^2 x\right) = \frac{\cos\left(\frac{\pi}{2} \sin^2 x\right)}{\sin\left(\frac{\pi}{2} \sin^2 x\right)} \]

### Step 2: Substitute into the limit
Now, we can substitute this back into our limit:
\[ \lim_{x \to \frac{\pi}{2}} \frac{\cos\left(\frac{\pi}{2} \sin^2 x\right)}{\frac{\cos\left(\frac{\pi}{2} \sin^2 x\right)}{\sin\left(\frac{\pi}{2} \sin^2 x\right)}} \]
This simplifies to:
\[ \lim_{x \to \frac{\pi}{2}} \sin\left(\frac{\pi}{2} \sin^2 x\right) \]

### Step 3: Evaluate \(\sin^2 x\) as \(x\) approaches \(\frac{\pi}{2}\)
As \(x\) approaches \(\frac{\pi}{2}\), we have:
\[ \sin^2 x \to \sin^2\left(\frac{\pi}{2}\right) = 1 \]

### Step 4: Substitute into the sine function
Now substituting this into the sine function:
\[ \sin\left(\frac{\pi}{2} \cdot 1\right) = \sin\left(\frac{\pi}{2}\right) = 1 \]

### Step 5: Conclusion
Thus, the limit evaluates to:
\[ \lim_{x \to \frac{\pi}{2}} \frac{\cos\left(\frac{\pi}{2} \sin^2 x\right)}{\cot\left(\frac{\pi}{2} \sin^2 x\right)} = 1 \]

### Final Answer
The value of the limit is \(1\).

---