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 follow these steps:

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

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

From the definition of the cosine function, we know:
\[ \cos(\theta) = \frac{1}{2} \]

Using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
we can find \( \sin^2(\theta) \):
\[ \sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4} \]

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

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

From the definition of the sine function, we know:
\[ \sin(\phi) = \frac{1}{3} \]

Using the Pythagorean identity again:
\[ \sin^2(\phi) + \cos^2(\phi) = 1 \]
we can find \( \cos^2(\phi) \):
\[ \cos^2(\phi) = 1 - \sin^2(\phi) = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]

### Step 3: Combine the results

Now, we can substitute the values we found back into the original expression:
\[ \sin^2(\cos^{-1}(1/2)) + \cos^2(\sin^{-1}(1/3)) = \frac{3}{4} + \frac{8}{9} \]

To add these fractions, we need a common denominator. The least common multiple of 4 and 9 is 36.

Convert each fraction:
\[ \frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36} \]
\[ \frac{8}{9} = \frac{8 \times 4}{9 \times 4} = \frac{32}{36} \]

Now add them:
\[ \frac{27}{36} + \frac{32}{36} = \frac{27 + 32}{36} = \frac{59}{36} \]

### Final Result

Thus, the value of \( \sin^2(\cos^{-1}(1/2)) + \cos^2(\sin^{-1}(1/3)) \) is:
\[ \frac{59}{36} \]

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