Answer:
The value of cos(A + B) in simplest form is:
cos(A + B) = 32/185 - sqrt(219465/246825)
Explanation:
To find the value of cos(A + B), we can use the trigonometric identity:
cos(A + B) = cosA * cosB - sinA * sinB
Given that tanA = 8/15, we can find sinA using the Pythagorean identity:
sinA = sqrt(1 - cos^2A)
Since tanA = sinA / cosA, we can substitute the values:
8/15 = sinA / cosA
Using the Pythagorean identity, we can solve for sinA:
sinA = sqrt(1 - (8/15)^2) = sqrt(1 - 64/225) = sqrt(161/225)
Similarly, we can find sinB using the Pythagorean identity:
sinB = sqrt(1 - cos^2B) = sqrt(1 - (12/37)^2) = sqrt(1365/1369)
Now we have the values of sinA, cosA, sinB, and cosB. We can substitute them into the formula for cos(A + B):
cos(A + B) = cosA * cosB - sinA * sinB
cos(A + B) = (8/15) * (12/37) - (sqrt(161/225)) * (sqrt(1365/1369))
cos(A + B) = 96/555 - sqrt(161/225 * 1365/1369)
Simplifying the expression further, we have:
cos(A + B) = 32/185 - sqrt(161 * 1365 / 225 * 1369)
Thus, the value of cos(A + B) in simplest form is:
cos(A + B) = 32/185 - sqrt(219465/246825)