The number of roots of `cos^(-1)x+cos^(-1)2x= pi/2` is

Options

A. 0
B. 1
C. 2
D. `gt2`

Answer

To solve the equation \( \cos^{-1} x + \cos^{-1} 2x = \frac{\pi}{2} \), we will follow these steps:

### Step 1: Understand the domain of the functions involved
The function \( \cos^{-1} x \) is defined for \( x \) in the interval \([-1, 1]\). For \( \cos^{-1} 2x \) to be defined, \( 2x \) must also lie within this interval. Therefore, we need:
\[ -1 \leq 2x \leq 1 \]
This simplifies to:
\[ -\frac{1}{2} \leq x \leq \frac{1}{2} \]
So, the domain of \( x \) is \([-0.5, 0.5]\).

### Step 2: Rewrite the equation
We can rewrite the equation as:
\[ \cos^{-1} x = \frac{\pi}{2} - \cos^{-1} 2x \]
Using the identity \( \cos^{-1} a + \cos^{-1} b = \frac{\pi}{2} \) when \( a^2 + b^2 = 1 \), we can set:
\[ x^2 + (2x)^2 = 1 \]

### Step 3: Solve the equation
Expanding the equation gives:
\[ x^2 + 4x^2 = 1 \implies 5x^2 = 1 \implies x^2 = \frac{1}{5} \implies x = \pm \frac{1}{\sqrt{5}} \]

### Step 4: Check if the solutions are within the domain
Now we need to check if these solutions fall within the interval \([-0.5, 0.5]\):
\[ -\frac{1}{\sqrt{5}} \approx -0.447 \quad \text{and} \quad \frac{1}{\sqrt{5}} \approx 0.447 \]
Both values are within the interval \([-0.5, 0.5]\).

### Step 5: Count the number of roots
Since both \( x = -\frac{1}{\sqrt{5}} \) and \( x = \frac{1}{\sqrt{5}} \) are valid solutions within the domain, we conclude that there are **2 roots** for the equation.

### Final Answer
The number of roots of the equation \( \cos^{-1} x + \cos^{-1} 2x = \frac{\pi}{2} \) is **2**.

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