Evaluate the following :
`sin ^(2) ( cos ^(-1) . 1/2) + cos^(2) ( sin^(-1) . 1/3) ` .

Answer

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

### Step 1: Evaluate \( \cos^{-1}(1/2) \)
Let \( A = \cos^{-1}(1/2) \). This means \( \cos A = 1/2 \). We know that:
\[ \cos 60^\circ = \frac{1}{2} \]
Thus, \( A = 60^\circ \).

### Step 2: Find \( \sin^2(\cos^{-1}(1/2)) \)
Using the identity \( \sin^2 A + \cos^2 A = 1 \), we can find \( \sin A \):
\[ \sin A = \sqrt{1 - \cos^2 A} = \sqrt{1 - (1/2)^2} = \sqrt{1 - 1/4} = \sqrt{3/4} = \frac{\sqrt{3}}{2} \]
Now, we calculate \( \sin^2 A \):
\[ \sin^2 A = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

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

### Step 4: Find \( \cos^2(\sin^{-1}(1/3)) \)
Using the identity \( \sin^2 B + \cos^2 B = 1 \), we can find \( \cos B \):
\[ \cos B = \sqrt{1 - \sin^2 B} = \sqrt{1 - (1/3)^2} = \sqrt{1 - 1/9} = \sqrt{8/9} = \frac{\sqrt{8}}{3} \]
Now, we calculate \( \cos^2 B \):
\[ \cos^2 B = \left(\frac{\sqrt{8}}{3}\right)^2 = \frac{8}{9} \]

### Step 5: Combine the results
Now we can combine the results from Step 2 and Step 4:
\[ \sin^2(\cos^{-1}(1/2)) + \cos^2(\sin^{-1}(1/3)) = \frac{3}{4} + \frac{8}{9} \]

### Step 6: Find a common denominator and add
The common denominator of 4 and 9 is 36. We convert each fraction:
\[ \frac{3}{4} = \frac{27}{36}, \quad \frac{8}{9} = \frac{32}{36} \]
Now, we add them:
\[ \frac{27}{36} + \frac{32}{36} = \frac{59}{36} \]

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

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