`cot^(-1)[(cos alpha)^(1//2)]-tan^(-1)[(cos alpha)^(1//2)]=x` then `sin x`=

Options

A. `tan^(2) (alpha//2)`
B. `cot^(2)(alpha//2)`
C. `tan alpha`
D. `cot alpha`

Answer

To solve the equation \( \cot^{-1}[(\cos \alpha)^{1/2}] - \tan^{-1}[(\cos \alpha)^{1/2}] = x \) and find \( \sin x \), we will follow these steps:

### Step 1: Rewrite the equation
We can express \( (\cos \alpha)^{1/2} \) as \( \sqrt{\cos \alpha} \). Thus, we rewrite the equation as:
\[ \cot^{-1}(\sqrt{\cos \alpha}) - \tan^{-1}(\sqrt{\cos \alpha}) = x \]

### Step 2: Use the cotangent and tangent identity
We know from trigonometric identities that:
\[ \cot^{-1}(y) + \tan^{-1}(y) = \frac{\pi}{2} \]
for any \( y \). Therefore, we can rewrite the equation as:
\[ \cot^{-1}(\sqrt{\cos \alpha}) + \tan^{-1}(\sqrt{\cos \alpha}) = \frac{\pi}{2} \]

### Step 3: Add the two equations
Adding the two equations gives:
\[ \cot^{-1}(\sqrt{\cos \alpha}) + \tan^{-1}(\sqrt{\cos \alpha}) - \tan^{-1}(\sqrt{\cos \alpha}) = x + \frac{\pi}{2} \]
This simplifies to:
\[ \cot^{-1}(\sqrt{\cos \alpha}) = x + \frac{\pi}{2} \]

### Step 4: Solve for \( x \)
From the above equation, we can express \( x \) as:
\[ x = \cot^{-1}(\sqrt{\cos \alpha}) - \frac{\pi}{2} \]

### Step 5: Use the cotangent identity
Using the identity for cotangent, we can express \( \cot^{-1}(y) \) in terms of sine:
\[ \cot^{-1}(y) = \frac{\pi}{2} - \tan^{-1}(y) \]
Thus, we can rewrite \( x \):
\[ x = -\tan^{-1}(\sqrt{\cos \alpha}) \]

### Step 6: Find \( \sin x \)
Using the sine of an angle:
\[ \sin(-\theta) = -\sin(\theta) \]
we have:
\[ \sin x = \sin(-\tan^{-1}(\sqrt{\cos \alpha})) = -\sin(\tan^{-1}(\sqrt{\cos \alpha})) \]

### Step 7: Use the sine identity
From the definition of sine in terms of tangent:
\[ \sin(\tan^{-1}(y)) = \frac{y}{\sqrt{1+y^2}} \]
we can substitute \( y = \sqrt{\cos \alpha} \):
\[ \sin(\tan^{-1}(\sqrt{\cos \alpha})) = \frac{\sqrt{\cos \alpha}}{\sqrt{1 + \cos \alpha}} \]

### Step 8: Final expression for \( \sin x \)
Thus, we find:
\[ \sin x = -\frac{\sqrt{\cos \alpha}}{\sqrt{1 + \cos \alpha}} \]

### Conclusion
The final answer is:
\[ \sin x = -\frac{\sqrt{\cos \alpha}}{\sqrt{1 + \cos \alpha}} \]