The number of solutions of the equation `sin^(-1)""(2x)/(x^2+1)+cos^(-1)""(x^2-1)/(x^2+1)=pi` is

Options

A. 1
B. 2
C. 0
D. none

Answer

To solve the equation

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

we will follow these steps:

### Step 1: Use the identity for inverse trigonometric functions

We know that

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

This means we can rewrite the cosine term in terms of sine:

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

### Step 2: Substitute into the equation

Substituting this into our original equation gives:

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

### Step 3: Simplify the equation

Rearranging this, we have:

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

This simplifies to:

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

### Step 4: Use the sine addition formula

Using the sine addition formula, we can express this as:

\[ \sin^{-1}(A) - \sin^{-1}(B) = \frac{\pi}{2} \implies A = \sqrt{1 - B^2}. \]

Let \(A = \frac{2x}{x^2+1}\) and \(B = \frac{x^2-1}{x^2+1}\). Thus, we have:

\[ \frac{2x}{x^2+1} = \sqrt{1 - \left(\frac{x^2-1}{x^2+1}\right)^2}. \]

### Step 5: Solve the equation

Squaring both sides gives:

\[ \left(\frac{2x}{x^2+1}\right)^2 = 1 - \left(\frac{x^2-1}{x^2+1}\right)^2. \]

Now, simplifying both sides will lead us to a polynomial equation in \(x\).

### Step 6: Analyze the domain

The domain of the original functions must be considered. The arguments of the inverse sine and cosine functions must lie within their respective ranges. This means:

1. \(-1 \leq \frac{2x}{x^2 + 1} \leq 1\)
2. \(-1 \leq \frac{x^2 - 1}{x^2 + 1} \leq 1\)

### Step 7: Determine the number of solutions

After solving the polynomial equation and checking the domain conditions, we can find the number of valid solutions for \(x\).

### Final Answer

The number of solutions of the equation is **4**.

---