If `cos^(-1)"" 3/5 - sin^(-1) ""4/5 = cos^(-1) x, " then " x=`

Options

A. 0
B. 1
C. `1/2`
D. `1/4`

Answer

To solve the equation \( \cos^{-1} \left( \frac{3}{5} \right) - \sin^{-1} \left( \frac{4}{5} \right) = \cos^{-1} x \), we will follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ \cos^{-1} \left( \frac{3}{5} \right) - \sin^{-1} \left( \frac{4}{5} \right) = \cos^{-1} x \]

### Step 2: Use the identity for sine and cosine
We know that:
\[ \sin^{-1} y + \cos^{-1} y = \frac{\pi}{2} \]
From this, we can express \( \sin^{-1} \left( \frac{4}{5} \right) \) in terms of cosine:
\[ \sin^{-1} \left( \frac{4}{5} \right) = \frac{\pi}{2} - \cos^{-1} \left( \frac{4}{5} \right) \]

### Step 3: Substitute the identity into the equation
Substituting this into our equation gives:
\[ \cos^{-1} \left( \frac{3}{5} \right) - \left( \frac{\pi}{2} - \cos^{-1} \left( \frac{4}{5} \right) \right) = \cos^{-1} x \]
This simplifies to:
\[ \cos^{-1} \left( \frac{3}{5} \right) + \cos^{-1} \left( \frac{4}{5} \right) - \frac{\pi}{2} = \cos^{-1} x \]

### Step 4: Use the cosine addition formula
Using the cosine addition formula:
\[ \cos^{-1} a + \cos^{-1} b = \cos^{-1} (ab - \sqrt{(1-a^2)(1-b^2)}) \]
for \( a = \frac{3}{5} \) and \( b = \frac{4}{5} \):
\[ \cos^{-1} \left( \frac{3}{5} \right) + \cos^{-1} \left( \frac{4}{5} \right) = \cos^{-1} \left( \frac{3}{5} \cdot \frac{4}{5} - \sqrt{\left(1 - \left(\frac{3}{5}\right)^2\right)\left(1 - \left(\frac{4}{5}\right)^2\right)} \right) \]

### Step 5: Calculate the components
Calculating \( ab \):
\[ ab = \frac{3}{5} \cdot \frac{4}{5} = \frac{12}{25} \]
Calculating \( 1 - a^2 \) and \( 1 - b^2 \):
\[ 1 - \left( \frac{3}{5} \right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
\[ 1 - \left( \frac{4}{5} \right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]
Now, calculate the square root:
\[ \sqrt{\left(1 - \left(\frac{3}{5}\right)^2\right)\left(1 - \left(\frac{4}{5}\right)^2\right)} = \sqrt{\frac{16}{25} \cdot \frac{9}{25}} = \sqrt{\frac{144}{625}} = \frac{12}{25} \]

### Step 6: Substitute back
Now substituting back:
\[ \cos^{-1} \left( \frac{12}{25} - \frac{12}{25} \right) = \cos^{-1} 0 \]
Thus, we have:
\[ \cos^{-1} 0 = \frac{\pi}{2} \]

### Step 7: Solve for x
From the equation:
\[ \cos^{-1} 0 = \cos^{-1} x \implies x = 1 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{1} \]