Consider the following two statements P and Q.
`P: cos^(-1)(cos.(4pi)/(3))=(4pi)/(3)`
`Q:sec^(2)(cot^(-1).(1)/(2))+"cosec"^(2)(tan^(-1).(1)/(3))=15`
Then, which of the following true?

Options

A. Both P and Q are true
B. P is true, but Q is false
C. P is false, but Q is true
D. Both P and Q are false

Answer

To determine the truth of the statements P and Q, we will analyze each statement step by step.

### Step 1: Analyze Statement P
**Statement P:** \( \cos^{-1}(\cos(4\pi/3)) = 4\pi/3 \)

1. **Understanding the range of \( \cos^{-1} \)**:
The function \( \cos^{-1}(x) \) returns values in the range \( [0, \pi] \).

2. **Calculate \( \cos(4\pi/3) \)**:
\[ \cos(4\pi/3) = \cos(\pi + \pi/3) = -\cos(\pi/3) = -\frac{1}{2} \]

3. **Evaluate \( \cos^{-1}(-\frac{1}{2}) \)**:
The angle whose cosine is \(-\frac{1}{2}\) in the range \( [0, \pi] \) is:
\[ \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \]

4. **Compare with \( 4\pi/3 \)**:
Since \( \frac{2\pi}{3} \neq \frac{4\pi}{3} \), we conclude:
\[ P \text{ is false.} \]

### Step 2: Analyze Statement Q
**Statement Q:** \( \sec^2(\cot^{-1}(1/2)) + \csc^2(\tan^{-1}(1/3)) = 15 \)

1. **Evaluate \( \sec^2(\cot^{-1}(1/2)) \)**:
Let \( \theta = \cot^{-1}(1/2) \). Then, \( \cot(\theta) = 1/2 \).
Using the identity \( \sec^2(\theta) = 1 + \cot^2(\theta) \):
\[ \sec^2(\theta) = 1 + \left(\frac{1}{2}\right)^2 = 1 + \frac{1}{4} = \frac{5}{4} \]

2. **Evaluate \( \csc^2(\tan^{-1}(1/3)) \)**:
Let \( \phi = \tan^{-1}(1/3) \). Then, \( \tan(\phi) = 1/3 \).
Using the identity \( \csc^2(\phi) = 1 + \tan^2(\phi) \):
\[ \csc^2(\phi) = 1 + \left(\frac{1}{3}\right)^2 = 1 + \frac{1}{9} = \frac{10}{9} \]

3. **Combine the results**:
Now, we add the two results:
\[ \sec^2(\cot^{-1}(1/2)) + \csc^2(\tan^{-1}(1/3)) = \frac{5}{4} + \frac{10}{9} \]

To add these fractions, we need a common denominator, which is \( 36 \):
\[ \frac{5}{4} = \frac{5 \times 9}{4 \times 9} = \frac{45}{36} \]
\[ \frac{10}{9} = \frac{10 \times 4}{9 \times 4} = \frac{40}{36} \]
\[ \frac{45}{36} + \frac{40}{36} = \frac{85}{36} \]

4. **Compare with 15**:
Since \( \frac{85}{36} \neq 15 \), we conclude:
\[ Q \text{ is false.} \]

### Conclusion
Both statements P and Q are false.