Evaluate the following expression:
`cos("sin"^(-1)4/5+"cos"^(-1)2/3)`

Explanation

To evaluate the expression \( \cos(\sin^{-1}(4/5) + \cos^{-1}(2/3)) \), we can follow these steps:

### Step 1: Define Variables
Let:
- \( a = \sin^{-1}(4/5) \)
- \( b = \cos^{-1}(2/3) \)

### Step 2: Find Sine and Cosine Values
From the definitions:
- \( \sin(a) = 4/5 \)
- \( \cos(b) = 2/3 \)

### Step 3: Calculate \( \cos(a) \)
Using the Pythagorean identity:
\[ \cos(a) = \sqrt{1 - \sin^2(a)} = \sqrt{1 - (4/5)^2} \]
Calculating \( (4/5)^2 \):
\[ (4/5)^2 = 16/25 \]
Thus,
\[ \cos(a) = \sqrt{1 - 16/25} = \sqrt{9/25} = \frac{3}{5} \]

### Step 4: Calculate \( \sin(b) \)
Using the Pythagorean identity:
\[ \sin(b) = \sqrt{1 - \cos^2(b)} = \sqrt{1 - (2/3)^2} \]
Calculating \( (2/3)^2 \):
\[ (2/3)^2 = 4/9 \]
Thus,
\[ \sin(b) = \sqrt{1 - 4/9} = \sqrt{5/9} = \frac{\sqrt{5}}{3} \]

### Step 5: Use the Cosine Addition Formula
Now we need to find \( \cos(a + b) \) using the formula:
\[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \]
Substituting the values we found:
\[ \cos(a + b) = \left(\frac{3}{5}\right)\left(\frac{2}{3}\right) - \left(\frac{4}{5}\right)\left(\frac{\sqrt{5}}{3}\right) \]

### Step 6: Simplify the Expression
Calculating each term:
1. \( \cos(a)\cos(b) = \frac{3}{5} \cdot \frac{2}{3} = \frac{2}{5} \)
2. \( \sin(a)\sin(b) = \frac{4}{5} \cdot \frac{\sqrt{5}}{3} = \frac{4\sqrt{5}}{15} \)

Now substituting back:
\[ \cos(a + b) = \frac{2}{5} - \frac{4\sqrt{5}}{15} \]

### Step 7: Find a Common Denominator
The common denominator between 5 and 15 is 15:
\[ \cos(a + b) = \frac{6}{15} - \frac{4\sqrt{5}}{15} = \frac{6 - 4\sqrt{5}}{15} \]

### Final Answer
Thus, the value of the expression \( \cos(\sin^{-1}(4/5) + \cos^{-1}(2/3)) \) is:
\[ \frac{6 - 4\sqrt{5}}{15} \]

---