If `log x = (-1)/3 , log y = 2/5 " and " P =log ( sin ( arc cos sqrt(1 - x^(2))))`
`Q = log (cos ( arc tan (sqrt(1-x^(2)y^(2))/(xy))))`, then

Options

A. a. `P = (-1)/9`
B. b. `P + Q = (-4)/15`
C. c. `P - Q = (-2)/5`
D. d. `P/Q = - 5`

Answer

To solve the given problem, we will follow these steps:

### Step 1: Find the values of x and y
Given:
- \(\log x = -\frac{1}{3}\)
- \(\log y = \frac{2}{5}\)

We can convert these logarithmic equations to exponential form:
- \(x = 10^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{10}}\)
- \(y = 10^{\frac{2}{5}} = 10^{0.4}\)

### Step 2: Calculate P
We need to find:
\[ P = \log(\sin(\arccos(\sqrt{1 - x^2}))) \]

Using the identity:
\[ \sin(\arccos(a)) = \sqrt{1 - a^2} \]
we can substitute \(a = \sqrt{1 - x^2}\):
\[ P = \log(\sqrt{1 - x^2}) \]

Now, we calculate \(x^2\):
\[ x^2 = \left(\frac{1}{\sqrt[3]{10}}\right)^2 = \frac{1}{10^{\frac{2}{3}}} \]
Thus,
\[ 1 - x^2 = 1 - \frac{1}{10^{\frac{2}{3}}} = \frac{10^{\frac{2}{3}} - 1}{10^{\frac{2}{3}}} \]
Now, substituting this into the expression for \(P\):
\[ P = \log\left(\sqrt{\frac{10^{\frac{2}{3}} - 1}{10^{\frac{2}{3}}}}\right) = \frac{1}{2} \log(10^{\frac{2}{3}} - 1) - \frac{1}{2} \log(10^{\frac{2}{3}}) \]

### Step 3: Calculate Q
Next, we find:
\[ Q = \log\left(\cos\left(\tan^{-1}\left(\frac{\sqrt{1 - x^2 y^2}}{xy}\right)\right)\right) \]

Using the identity:
\[ \cos(\tan^{-1}(b)) = \frac{1}{\sqrt{1 + b^2}} \]
where \(b = \frac{\sqrt{1 - x^2 y^2}}{xy}\):
\[ Q = \log\left(\frac{1}{\sqrt{1 + \left(\frac{\sqrt{1 - x^2 y^2}}{xy}\right)^2}}\right) \]
Calculating \(1 + b^2\):
\[ b^2 = \frac{1 - x^2 y^2}{x^2 y^2} \]
Thus,
\[ 1 + b^2 = \frac{x^2 y^2 + 1 - x^2 y^2}{x^2 y^2} = \frac{1}{x^2 y^2} \]
So,
\[ Q = \log\left(\sqrt{xy}\right) = \frac{1}{2} \log(xy) \]

### Step 4: Combine P and Q
Now we can find \(P + Q\):
\[ P + Q = \log\left(\sqrt{1 - x^2}\right) + \frac{1}{2} \log(xy) \]

### Step 5: Evaluate Options
Now we can evaluate the options based on the values of \(P\) and \(Q\):
- \(P = -\frac{1}{3}\)
- \(Q = \frac{1}{15}\)

1. **Option A**: \(P = -\frac{1}{3}\) (Correct)
2. **Option B**: \(P + Q = -\frac{1}{3} + \frac{1}{15} = -\frac{4}{15}\) (Correct)
3. **Option C**: \(P - Q = -\frac{1}{3} - \frac{1}{15} = -\frac{6}{15} = -\frac{2}{5}\) (Correct)
4. **Option D**: \(\frac{P}{Q} = \frac{-\frac{1}{3}}{\frac{1}{15}} = -5\) (Correct)

### Final Result
All options are correct.