If m and M are the least and the greatest value of `(cos^(-1)x)^(2)+(sin^(-1)x)^(2)` , then `(M)/(m)` is equal to

Options

A. 10
B. 5
C. 4
D. 2

Explanation

To solve the problem, we need to find the least value (m) and the greatest value (M) of the expression \((\cos^{-1} x)^2 + (\sin^{-1} x)^2\) and then compute \(\frac{M}{m}\).

### Step-by-Step Solution:

1. **Use the Identity**:
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
Therefore, we can express \(\cos^{-1} x\) in terms of \(\sin^{-1} x\):
\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]

2. **Substitute into the Expression**:
Substitute \(\cos^{-1} x\) into the original expression:
\[ (\cos^{-1} x)^2 + (\sin^{-1} x)^2 = \left(\frac{\pi}{2} - \sin^{-1} x\right)^2 + (\sin^{-1} x)^2 \]

3. **Expand the Square**:
Using the formula \((a - b)^2 = a^2 - 2ab + b^2\):
\[ = \left(\frac{\pi^2}{4} - \pi \sin^{-1} x + (\sin^{-1} x)^2\right) + (\sin^{-1} x)^2 \]
\[ = \frac{\pi^2}{4} - \pi \sin^{-1} x + 2(\sin^{-1} x)^2 \]

4. **Rearranging the Expression**:
Now we have:
\[ f(y) = 2y^2 - \pi y + \frac{\pi^2}{4} \]
where \(y = \sin^{-1} x\).

5. **Finding the Vertex**:
This is a quadratic function in \(y\). The vertex of a quadratic \(ay^2 + by + c\) occurs at:
\[ y = -\frac{b}{2a} = \frac{\pi}{4} \]
Substitute \(y = \frac{\pi}{4}\) back into the expression to find the minimum value \(m\):
\[ f\left(\frac{\pi}{4}\right) = 2\left(\frac{\pi}{4}\right)^2 - \pi\left(\frac{\pi}{4}\right) + \frac{\pi^2}{4} \]
\[ = 2 \cdot \frac{\pi^2}{16} - \frac{\pi^2}{4} + \frac{\pi^2}{4} = \frac{\pi^2}{8} \]

6. **Finding the Maximum Value**:
The maximum value occurs at the endpoints of the domain of \(\sin^{-1} x\), which is \([-1, 1]\). Thus, we evaluate \(f(-\frac{\pi}{2})\) and \(f(\frac{\pi}{2})\):
- For \(y = -\frac{\pi}{2}\):
\[ f\left(-\frac{\pi}{2}\right) = 2\left(-\frac{\pi}{2}\right)^2 - \pi\left(-\frac{\pi}{2}\right) + \frac{\pi^2}{4} \]
\[ = 2 \cdot \frac{\pi^2}{4} + \frac{\pi^2}{2} + \frac{\pi^2}{4} = \frac{2\pi^2}{4} + \frac{2\pi^2}{4} = \frac{4\pi^2}{4} = \pi^2 \]
- For \(y = \frac{\pi}{2}\):
\[ f\left(\frac{\pi}{2}\right) = 2\left(\frac{\pi}{2}\right)^2 - \pi\left(\frac{\pi}{2}\right) + \frac{\pi^2}{4} \]
\[ = 2 \cdot \frac{\pi^2}{4} - \frac{\pi^2}{2} + \frac{\pi^2}{4} = \frac{2\pi^2}{4} - \frac{2\pi^2}{4} + \frac{\pi^2}{4} = \frac{\pi^2}{4} \]

The maximum value \(M = \pi^2\).

7. **Calculating \(\frac{M}{m}\)**:
\[ \frac{M}{m} = \frac{\pi^2}{\frac{\pi^2}{8}} = 8 \]

### Final Answer:
\[ \frac{M}{m} = 8 \]