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.

### Statement P:
**P: cos^(-1)(cos(4π/3)) = (4π/3)**

1. **Understanding the function**: The function cos^(-1)(x) gives the angle whose cosine is x, and it is defined in the range [0, π].

2. **Calculate cos(4π/3)**:
- The angle 4π/3 is in the third quadrant where cosine is negative.
- cos(4π/3) = -1/2.

3. **Using the property of cos^(-1)**:
- cos^(-1)(cos(θ)) = θ if θ is in [0, π].
- Since 4π/3 is not in [0, π], we need to find an equivalent angle in this range.
- We can express 4π/3 as:
\[ 4π/3 = 2π - 2π/3 \]
- The equivalent angle in [0, π] is:
\[ 2π/3 \]

4. **Final calculation**:
- Therefore, cos^(-1)(cos(4π/3)) = cos^(-1)(-1/2) = 2π/3.
- Since 2π/3 ≠ 4π/3, statement P is **false**.

### Statement Q:
**Q: sec^2(cot^(-1)(1/2)) + cosec^2(tan^(-1)(1/3)) = 15**

1. **Calculate sec^2(cot^(-1)(1/2))**:
- Let θ = cot^(-1)(1/2). This means cot(θ) = 1/2.
- In a right triangle, if cot(θ) = 1/2, then the opposite side is 1 and the adjacent side is 2.
- The hypotenuse can be calculated using the Pythagorean theorem:
\[ hypotenuse = \sqrt{1^2 + 2^2} = \sqrt{5} \]
- Therefore, sec(θ) = hypotenuse/adjacent = √5/2.
- Thus, sec^2(θ) = (√5/2)^2 = 5/4.

2. **Calculate cosec^2(tan^(-1)(1/3))**:
- Let φ = tan^(-1)(1/3). This means tan(φ) = 1/3.
- In a right triangle, if tan(φ) = 1/3, then the opposite side is 1 and the adjacent side is 3.
- The hypotenuse can be calculated using the Pythagorean theorem:
\[ hypotenuse = \sqrt{1^2 + 3^2} = \sqrt{10} \]
- Therefore, cosec(φ) = hypotenuse/opposite = √10/1.
- Thus, cosec^2(φ) = (√10)^2 = 10.

3. **Final calculation**:
- Now we can sum the two results:
\[ sec^2(cot^(-1)(1/2)) + cosec^2(tan^(-1)(1/3)) = \frac{5}{4} + 10 = \frac{5}{4} + \frac{40}{4} = \frac{45}{4} \]
- Since 45/4 = 11.25, which is not equal to 15, statement Q is also **false**.

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