The value of `sin^2(cos^(-1)(1/2) )+ cos^2(sin^(-1)(1/3))` is: (A) `17/36` (B) `59/36` (C) `36/59` (D) none of these

Options

A. `(17)/(36)`
B. `(59)/(36)`
C. `(36)/(59)`
D. None of these

Answer

To solve the expression \( \sin^2(\cos^{-1}(1/2)) + \cos^2(\sin^{-1}(1/3)) \), we will follow these steps:

### Step 1: Evaluate \( \cos^{-1}(1/2) \)
The value of \( \cos^{-1}(1/2) \) corresponds to the angle whose cosine is \( 1/2 \). This angle is \( \frac{\pi}{3} \) or \( 60^\circ \).

### Step 2: Find \( \sin(\cos^{-1}(1/2)) \)
Using the identity \( \sin(\theta) = \sqrt{1 - \cos^2(\theta)} \):
\[ \sin(\cos^{-1}(1/2)) = \sqrt{1 - (1/2)^2} = \sqrt{1 - 1/4} = \sqrt{3/4} = \frac{\sqrt{3}}{2} \]

### Step 3: Calculate \( \sin^2(\cos^{-1}(1/2)) \)
Now, we square the result:
\[ \sin^2(\cos^{-1}(1/2)) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 4: Evaluate \( \sin^{-1}(1/3) \)
Let \( \theta = \sin^{-1}(1/3) \). This means \( \sin(\theta) = 1/3 \).

### Step 5: Find \( \cos(\sin^{-1}(1/3)) \)
Using the identity \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \):
\[ \cos(\sin^{-1}(1/3)) = \sqrt{1 - (1/3)^2} = \sqrt{1 - 1/9} = \sqrt{8/9} = \frac{2\sqrt{2}}{3} \]

### Step 6: Calculate \( \cos^2(\sin^{-1}(1/3)) \)
Now, we square the result:
\[ \cos^2(\sin^{-1}(1/3)) = \left(\frac{2\sqrt{2}}{3}\right)^2 = \frac{8}{9} \]

### Step 7: Combine the results
Now, we combine the two parts:
\[ \sin^2(\cos^{-1}(1/2)) + \cos^2(\sin^{-1}(1/3)) = \frac{3}{4} + \frac{8}{9} \]

### Step 8: Find a common denominator
The least common multiple of 4 and 9 is 36. We convert both fractions:
\[ \frac{3}{4} = \frac{27}{36}, \quad \frac{8}{9} = \frac{32}{36} \]

### Step 9: Add the fractions
Now we add the two fractions:
\[ \frac{27}{36} + \frac{32}{36} = \frac{59}{36} \]

### Final Answer
Thus, the value of \( \sin^2(\cos^{-1}(1/2)) + \cos^2(\sin^{-1}(1/3)) \) is \( \frac{59}{36} \).