if `cos^(- 1)sqrt(p)+cos^(- 1)sqrt(1-p)+cos^(- 1)sqrt(1-q)=(3pi)/4` ,then the value of q is (A) 1 (B) `1/sqrt(2)` (C) `1/3` (D) `1/2`

Options

A. 1
B. `(1)/(sqrt2)`
C. `(1)/(3)`
D. `(1)/(2)`

Explanation

To solve the equation

\[ \cos^{-1}(\sqrt{p}) + \cos^{-1}(\sqrt{1-p}) + \cos^{-1}(\sqrt{1-q}) = \frac{3\pi}{4}, \]

we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:

\[ \cos^{-1}(\sqrt{p}) + \cos^{-1}(\sqrt{1-p}) + \cos^{-1}(\sqrt{1-q}) = \frac{3\pi}{4}. \]

### Step 2: Use the identity for inverse cosine and sine
Using the identity that states

\[ \cos^{-1}(x) + \sin^{-1}(x) = \frac{\pi}{2}, \]

we can rewrite \(\cos^{-1}(\sqrt{p}) + \cos^{-1}(\sqrt{1-p})\) as follows:

\[ \cos^{-1}(\sqrt{p}) + \sin^{-1}(\sqrt{p}) = \frac{\pi}{2}. \]

Thus, we can express \(\cos^{-1}(\sqrt{1-p})\) as:

\[ \cos^{-1}(\sqrt{1-p}) = \frac{\pi}{2} - \sin^{-1}(\sqrt{p}). \]

### Step 3: Substitute back into the equation
Now substituting this back into the equation gives:

\[ \frac{\pi}{2} + \cos^{-1}(\sqrt{1-q}) = \frac{3\pi}{4}. \]

### Step 4: Isolate \(\cos^{-1}(\sqrt{1-q})\)
Subtract \(\frac{\pi}{2}\) from both sides:

\[ \cos^{-1}(\sqrt{1-q}) = \frac{3\pi}{4} - \frac{\pi}{2}. \]

Calculating the right-hand side:

\[ \frac{3\pi}{4} - \frac{2\pi}{4} = \frac{\pi}{4}. \]

### Step 5: Remove the inverse cosine
Now we have:

\[ \cos^{-1}(\sqrt{1-q}) = \frac{\pi}{4}. \]

Taking the cosine of both sides gives:

\[ \sqrt{1-q} = \cos\left(\frac{\pi}{4}\right). \]

Since \(\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), we have:

\[ \sqrt{1-q} = \frac{1}{\sqrt{2}}. \]

### Step 6: Square both sides
Squaring both sides results in:

\[ 1 - q = \frac{1}{2}. \]

### Step 7: Solve for \(q\)
Rearranging gives:

\[ q = 1 - \frac{1}{2} = \frac{1}{2}. \]

Thus, the value of \(q\) is

\[ \boxed{\frac{1}{2}}. \]