Let the slope of the curve `y=cos^(-1)(sinx)` be `tantheta` then the value of `theta` in the interval `(0,pi)` is

Options

A. `(pi)/(6)`
B. `(3pi)/(4)`
C. `(pi)/(4)`
D. `(pi)/(2)`

Answer

To solve the problem, we need to find the value of \( \theta \) such that the slope of the curve \( y = \cos^{-1}(\sin x) \) is equal to \( \tan \theta \). Here’s a step-by-step solution:

### Step 1: Rewrite the function
We start with the function given:
\[ y = \cos^{-1}(\sin x) \]
Using the identity \( \cos^{-1}(u) = \frac{\pi}{2} - \sin^{-1}(u) \), we can rewrite the function:
\[ y = \frac{\pi}{2} - \sin^{-1}(\sin x) \]
Since \( \sin^{-1}(\sin x) = x \) for \( x \) in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we can express \( y \) as:
\[ y = \frac{\pi}{2} - x \]

### Step 2: Differentiate the function
Next, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 0 - 1 = -1 \]

### Step 3: Relate the derivative to the slope
We know that the slope of the curve is given as \( \tan \theta \). Therefore, we can set up the equation:
\[ \frac{dy}{dx} = \tan \theta \]
Substituting the value we found:
\[ -1 = \tan \theta \]

### Step 4: Solve for \( \theta \)
To find \( \theta \), we need to determine the angle whose tangent is \(-1\). The tangent function is negative in the second and fourth quadrants. In the interval \( (0, \pi) \), the angle that satisfies this condition is:
\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Conclusion
Thus, the value of \( \theta \) in the interval \( (0, \pi) \) is:
\[ \theta = \frac{3\pi}{4} \]