The value of `sin^2("cos"^(-1)1/2)+cos^2("sin"^(-1)1/3)` is:

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 break down the problem step by step.

### Step 1: Find \( \sin^2(\cos^{-1}(1/2)) \)

Let \( \theta = \cos^{-1}(1/2) \). This means that \( \cos(\theta) = 1/2 \).

Using the Pythagorean identity, we can find \( \sin(\theta) \):
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Substituting \( \cos(\theta) = 1/2 \):
\[ \sin^2(\theta) + (1/2)^2 = 1 \]
\[ \sin^2(\theta) + 1/4 = 1 \]
\[ \sin^2(\theta) = 1 - 1/4 = 3/4 \]

Thus,
\[ \sin^2(\cos^{-1}(1/2)) = \frac{3}{4} \]

### Step 2: Find \( \cos^2(\sin^{-1}(1/3)) \)

Let \( \phi = \sin^{-1}(1/3) \). This means that \( \sin(\phi) = 1/3 \).

Using the Pythagorean identity again, we can find \( \cos(\phi) \):
\[ \sin^2(\phi) + \cos^2(\phi) = 1 \]
Substituting \( \sin(\phi) = 1/3 \):
\[ (1/3)^2 + \cos^2(\phi) = 1 \]
\[ 1/9 + \cos^2(\phi) = 1 \]
\[ \cos^2(\phi) = 1 - 1/9 = 8/9 \]

Thus,
\[ \cos^2(\sin^{-1}(1/3)) = \frac{8}{9} \]

### Step 3: Combine the results

Now, we can combine the results from Step 1 and Step 2:
\[ \sin^2(\cos^{-1}(1/2)) + \cos^2(\sin^{-1}(1/3)) = \frac{3}{4} + \frac{8}{9} \]

### Step 4: Find a common denominator and add

The least common multiple of 4 and 9 is 36. We convert both fractions:
\[ \frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36} \]
\[ \frac{8}{9} = \frac{8 \times 4}{9 \times 4} = \frac{32}{36} \]

Now, we can 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:
\[ \frac{59}{36} \]

---