The number of roots of the equation `sin^(-1)x-cos^(-1)x=sin^(-1)(5x-3)` is/ are

Options

A. 3
B. 1
C. 2
D. 0

Answer

To solve the equation \( \sin^{-1}x - \cos^{-1}x = \sin^{-1}(5x - 3) \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \cos^{-1}x = \frac{\pi}{2} - \sin^{-1}x \). Therefore, we can rewrite the left-hand side of the equation:

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

So the equation becomes:

\[ 2\sin^{-1}x - \frac{\pi}{2} = \sin^{-1}(5x - 3) \]

### Step 2: Isolate \( \sin^{-1}x \)
Rearranging gives us:

\[ 2\sin^{-1}x = \sin^{-1}(5x - 3) + \frac{\pi}{2} \]

### Step 3: Apply the sine function
Taking the sine of both sides, we have:

\[ \sin(2\sin^{-1}x) = \sin\left(\sin^{-1}(5x - 3) + \frac{\pi}{2}\right) \]

Using the sine addition formula, we know:

\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

Thus, we can write:

\[ \sin(2\sin^{-1}x) = \sin(\sin^{-1}(5x - 3)) \cos\left(\frac{\pi}{2}\right) + \cos(\sin^{-1}(5x - 3)) \sin\left(\frac{\pi}{2}\right) \]

Since \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \), this simplifies to:

\[ \sin(2\sin^{-1}x) = \cos(\sin^{-1}(5x - 3)) \]

### Step 4: Simplify further
Using the identity \( \sin(2\theta) = 2\sin\theta\cos\theta \):

\[ 2x\sqrt{1 - x^2} = \sqrt{1 - (5x - 3)^2} \]

### Step 5: Square both sides
Squaring both sides gives:

\[ (2x\sqrt{1 - x^2})^2 = (1 - (5x - 3)^2) \]

This leads to:

\[ 4x^2(1 - x^2) = 1 - (25x^2 - 30x + 9) \]

### Step 6: Expand and rearrange
Expanding both sides, we have:

\[ 4x^2 - 4x^4 = 1 - 25x^2 + 30x - 9 \]

Rearranging gives:

\[ 4x^4 - 29x^2 + 30x - 8 = 0 \]

### Step 7: Solve the polynomial
This is a quartic equation. We can use numerical methods or graphing to find the roots. However, we can also factor or use the quadratic formula on a reduced form if possible.

### Step 8: Check for valid roots
After finding the roots, we must check which of these roots are valid in the context of the original equation, particularly ensuring that the arguments of the inverse sine functions are within the domain of \([-1, 1]\).

### Conclusion
After checking the roots, we find that the only valid solution is \( x = \frac{1}{2} \). Therefore, the number of roots of the equation is:

\[ \text{Number of roots} = 1 \]

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