Number of solutions of the equation
`log_(10) ( sqrt(5 cos^(-1) x -1 )) + 1/2 log_(10) ( 2 cos^(-1) x + 3) + log_(10)sqrt5 = 1 ` is

Options

A. 0
B. 1
C. more than one but finite
D. infinite

Answer

To solve the equation

\[ \log_{10} \left( \sqrt{5 \cos^{-1} x - 1} \right) + \frac{1}{2} \log_{10} \left( 2 \cos^{-1} x + 3 \right) + \log_{10} \sqrt{5} = 1, \]

we will follow these steps:

### Step 1: Simplify the logarithmic expressions
Using the property of logarithms, \( \log_{10}(a^b) = b \log_{10}(a) \), we can rewrite the equation:

\[ \frac{1}{2} \log_{10} (5 \cos^{-1} x - 1) + \frac{1}{2} \log_{10} (2 \cos^{-1} x + 3) + \log_{10} \sqrt{5} = 1. \]

### Step 2: Combine the logarithms
Factor out \( \frac{1}{2} \) from the first two logarithmic terms:

\[ \frac{1}{2} \left( \log_{10} (5 \cos^{-1} x - 1) + \log_{10} (2 \cos^{-1} x + 3) \right) + \log_{10} \sqrt{5} = 1. \]

This can be rewritten as:

\[ \frac{1}{2} \log_{10} \left( (5 \cos^{-1} x - 1)(2 \cos^{-1} x + 3) \right) + \log_{10} \sqrt{5} = 1. \]

### Step 3: Eliminate the logarithm
Multiply both sides by 2:

\[ \log_{10} \left( (5 \cos^{-1} x - 1)(2 \cos^{-1} x + 3) \right) + 2 \log_{10} \sqrt{5} = 2. \]

Using the property \( \log_{10} a + \log_{10} b = \log_{10} (ab) \):

\[ \log_{10} \left( (5 \cos^{-1} x - 1)(2 \cos^{-1} x + 3) \cdot 5 \right) = 2. \]

### Step 4: Exponentiate to remove the logarithm
Exponentiate both sides:

\[ (5 \cos^{-1} x - 1)(2 \cos^{-1} x + 3) \cdot 5 = 10^2 = 100. \]

### Step 5: Rearrange the equation
This simplifies to:

\[ (5 \cos^{-1} x - 1)(2 \cos^{-1} x + 3) = 20. \]

### Step 6: Expand the left-hand side
Expanding gives us:

\[ 10 \cos^{-1} x - 2 + 15 \cos^{-1} x - 3 = 20. \]

Combining like terms results in:

\[ 10 \cos^{-1} x^2 + 13 \cos^{-1} x - 23 = 0. \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( \cos^{-1} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here \( a = 10, b = 13, c = -23 \):

\[ \cos^{-1} x = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 10 \cdot (-23)}}{2 \cdot 10}. \]

Calculating the discriminant:

\[ \sqrt{169 + 920} = \sqrt{1089} = 33. \]

Thus:

\[ \cos^{-1} x = \frac{-13 \pm 33}{20}. \]

Calculating the two potential solutions:

1. \( \cos^{-1} x = \frac{20}{20} = 1 \) implies \( x = \cos(1) \).
2. \( \cos^{-1} x = \frac{-46}{20} = -2.3 \) is not valid since \( \cos^{-1} x \) must be in the range \([0, \pi]\).

### Step 8: Conclusion
The only valid solution is \( x = \cos(1) \). Therefore, the number of solutions to the original equation is:

\[ \boxed{1}. \]