`cos("sin"^(-1)(1)/(2)+"cos"^(-1)(1)/(3))=`

Options

A. `(sqrt(3)+sqrt(8))/(6)`
B. `(-sqrt(3)+sqrt(8))/(6)`
C. `(sqrt(3)-sqrt(8))/(6)`
D. 0

Answer

To solve the problem \( \cos\left(\sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{1}{3}\right)\right) \), we can follow these steps:

### Step 1: Define the angles
Let \( \theta = \sin^{-1}\left(\frac{1}{2}\right) \). This means that \( \sin(\theta) = \frac{1}{2} \).

### Step 2: Create a right triangle
From the definition of sine, we can visualize this as a right triangle where:
- The opposite side (perpendicular) is 1 (since \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)).
- The hypotenuse is 2.

Using the Pythagorean theorem, we can find the adjacent side (base):
\[ \text{base} = \sqrt{\text{hypotenuse}^2 - \text{opposite}^2} = \sqrt{2^2 - 1^2} = \sqrt{4 - 1} = \sqrt{3}. \]

### Step 3: Find \( \cos(\theta) \)
Now we can find \( \cos(\theta) \):
\[ \cos(\theta) = \frac{\text{base}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}. \]

### Step 4: Define the second angle
Let \( \phi = \cos^{-1}\left(\frac{1}{3}\right) \). This means that \( \cos(\phi) = \frac{1}{3} \).

### Step 5: Use the cosine addition formula
We need to find \( \cos(\theta + \phi) \) using the cosine addition formula:
\[ \cos(\theta + \phi) = \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi). \]

### Step 6: Substitute values
We already have \( \cos(\theta) = \frac{\sqrt{3}}{2} \) and \( \sin(\theta) = \frac{1}{2} \). Now we need to find \( \sin(\phi) \).

Using \( \sin^2(\phi) + \cos^2(\phi) = 1 \):
\[ \sin^2(\phi) = 1 - \cos^2(\phi) = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}. \]
Thus,
\[ \sin(\phi) = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}. \]

### Step 7: Substitute into the cosine addition formula
Now we can substitute:
\[ \cos(\theta + \phi) = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{3}\right) - \left(\frac{1}{2}\right)\left(\frac{2\sqrt{2}}{3}\right). \]

### Step 8: Simplify the expression
Calculating each term:
\[ \cos(\theta + \phi) = \frac{\sqrt{3}}{6} - \frac{\sqrt{2}}{3}. \]
To combine these fractions, we can find a common denominator:
\[ \frac{\sqrt{3}}{6} - \frac{2\sqrt{2}}{6} = \frac{\sqrt{3} - 2\sqrt{2}}{6}. \]

### Final Answer
Thus, the final answer is:
\[ \cos\left(\sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{1}{3}\right)\right) = \frac{\sqrt{3} - 2\sqrt{2}}{6}. \]