Find the number of solutions of the equation `cos ^(1) (1-x) + m cos ^(-1) x = (npi)/(2),` where `m gt 0, n le 0.`

Answer

To find the number of solutions of the equation

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

where \( m > 0 \) and \( n \leq 0 \), we will analyze the equation step by step.

### Step 1: Understand the Domains and Ranges
The function \(\cos^{-1}(x)\) is defined for \(x\) in the interval \([-1, 1]\) and has a range of \([0, \pi]\). Therefore, we need to ensure that both \(1 - x\) and \(x\) fall within this domain.

**Hint:** Check the values of \(x\) that keep \(1 - x\) and \(x\) within the interval \([-1, 1]\).

### Step 2: Determine the Valid Range for \(x\)
From the condition \(1 - x \geq -1\), we get:

\[ 1 - x \geq -1 \implies x \leq 2 \]

From the condition \(1 - x \leq 1\), we have:

\[ 1 - x \leq 1 \implies x \geq 0 \]

From the condition \(x \geq -1\), we have:

\[ x \geq -1 \]

Combining these inequalities, we find that \(x\) must lie in the interval \([0, 1]\).

**Hint:** The values of \(x\) must satisfy both \(x \in [0, 1]\) and \(1 - x \in [0, 1]\).

### Step 3: Analyze the Equation
Now we rewrite the equation:

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

Given that \(m > 0\) and \(n \leq 0\), we know that \(\frac{n\pi}{2} \leq 0\).

**Hint:** Consider the implications of having a sum of two non-negative terms equal to a non-positive term.

### Step 4: Examine the Range of the Left Side
The left side, \(\cos^{-1}(1-x) + m \cos^{-1}(x)\), is the sum of two non-negative terms:

- \(\cos^{-1}(1-x)\) is \(0\) when \(x = 1\) and increases to \(\pi\) as \(x\) approaches \(0\).
- \(m \cos^{-1}(x)\) is \(0\) when \(x = 1\) and increases to \(m\pi/2\) as \(x\) approaches \(0\).

Thus, the left side ranges from \(0\) (when \(x = 1\)) to \(m\pi\) (when \(x = 0\)).

**Hint:** The left side must be non-negative, and since \(m > 0\), the left side cannot equal a negative number.

### Step 5: Conclusion
Since the left side of the equation is always non-negative and the right side is non-positive (as \(n \leq 0\)), the equation can never hold true. Therefore, there are no values of \(x\) that satisfy the equation.

Thus, the number of solutions to the equation is:

\[ \boxed{0} \]