The value of `cos^-1[cot(sin^-1(sqrt((2-sqrt3)/4))+cos^-1(sqrt12/4)+sec^-1sqrt2)`

Options

A. 0
B. `pi/4`
C. `pi/6`
D. `pi/2`

Answer

To solve the expression \( \cos^{-1} \left[ \cot \left( \sin^{-1} \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right) + \cos^{-1} \left( \frac{\sqrt{12}}{4} \right) + \sec^{-1} \sqrt{2} \right) \right] \), we will break it down step by step.

### Step 1: Simplify \( \sin^{-1} \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right) \)

Let \( x = \sin^{-1} \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right) \).

Using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ \sin x = \frac{\sqrt{2 - \sqrt{3}}}{2} \]
Thus,
\[ \cos x = \sqrt{1 - \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right)^2} \]
Calculating \( \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right)^2 \):
\[ \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right)^2 = \frac{2 - \sqrt{3}}{4} \]
So,
\[ 1 - \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right)^2 = 1 - \frac{2 - \sqrt{3}}{4} = \frac{4 - (2 - \sqrt{3})}{4} = \frac{2 + \sqrt{3}}{4} \]
Thus,
\[ \cos x = \frac{\sqrt{2 + \sqrt{3}}}{2} \]

### Step 2: Simplify \( \cos^{-1} \left( \frac{\sqrt{12}}{4} \right) \)

Calculating \( \frac{\sqrt{12}}{4} \):
\[ \frac{\sqrt{12}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6} \]

### Step 3: Simplify \( \sec^{-1} \sqrt{2} \)

Since \( \sec^{-1} x = \cos^{-1} \left( \frac{1}{x} \right) \):
\[ \sec^{-1} \sqrt{2} = \cos^{-1} \left( \frac{1}{\sqrt{2}} \right) = \frac{\pi}{4} \]

### Step 4: Combine the angles

Now we have:
\[ x + \frac{\pi}{6} + \frac{\pi}{4} \]
We need to find \( x \):
\[ x = \sin^{-1} \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right) \]
Using the identity for \( \sin^{-1} \):
\[ \sin^{-1} \left( \frac{\sqrt{2 - \sqrt{3}}}{2} \right) = \frac{\pi}{12} \]

Thus,
\[ \frac{\pi}{12} + \frac{\pi}{6} + \frac{\pi}{4} \]
Finding a common denominator (which is 12):
\[ \frac{\pi}{12} + \frac{2\pi}{12} + \frac{3\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \]

### Step 5: Find \( \cot \left( \frac{\pi}{2} \right) \)

\[ \cot \left( \frac{\pi}{2} \right) = 0 \]

### Step 6: Final Calculation

Now we need to find:
\[ \cos^{-1}(0) = \frac{\pi}{2} \]

Thus, the final answer is:
\[ \frac{\pi}{2} \]