If `cos^(-1)((7)/(|x|))+cos^(-1)((4sqrt(15))/(|x|))=(pi)/(2)`, then:

Options

A. `x= pm 16`
B. `x = pm 17`
C. `tan^(-1)(sqrt((|x|)/(|x|-12)))=(pi)/(3)`
D. `tan^(-1)((10|x|-1)/((|x|-4)^(2)))=(pi)/(4)`

Answer

To solve the equation

\[ \cos^{-1}\left(\frac{7}{|x|}\right) + \cos^{-1}\left(\frac{4\sqrt{15}}{|x|}\right) = \frac{\pi}{2}, \]

we can use the identity that states:

\[ \cos^{-1}(a) + \cos^{-1}(b) = \frac{\pi}{2} \implies a^2 + b^2 = 1. \]

### Step 1: Identify \( a \) and \( b \)

Let

\[ a = \frac{7}{|x|} \quad \text{and} \quad b = \frac{4\sqrt{15}}{|x|}. \]

### Step 2: Apply the identity

Using the identity, we have:

\[ \left(\frac{7}{|x|}\right)^2 + \left(\frac{4\sqrt{15}}{|x|}\right)^2 = 1. \]

### Step 3: Simplify the equation

Calculating the squares:

\[ \frac{49}{x^2} + \frac{240}{x^2} = 1. \]

Combining the fractions gives:

\[ \frac{49 + 240}{x^2} = 1. \]

### Step 4: Solve for \( x^2 \)

This simplifies to:

\[ \frac{289}{x^2} = 1. \]

Cross-multiplying gives:

\[ 289 = x^2. \]

### Step 5: Find \( x \)

Taking the square root of both sides, we find:

\[ x = \pm 17. \]

### Final Result

Thus, the solution to the equation is:

\[ x = 17 \quad \text{or} \quad x = -17. \]