find the exact value of the expression using the provided information. Find Tan (S+T) given that cos s=1/3 with s in quad I and sin T= -1/2 with T in quad IV

To find the value of tan(S + T), we need to use the trigonometric identities and the provided information about the values of cos(s) and sin(T).

1. Start by finding the value of sin(s) using the Pythagorean Identity: sin^2(s) + cos^2(s) = 1.

Given that cos(s) = 1/3, we can solve for sin(s):
sin^2(s) + (1/3)^2 = 1
sin^2(s) + 1/9 = 1
sin^2(s) = 1 - 1/9
sin^2(s) = 8/9

Since s is in quadrant I, sin(s) must be positive:
sin(s) = √(8/9) = √8/3

2. Next, find cos(T) using the Pythagorean Identity: sin^2(T) + cos^2(T) = 1.

Given that sin(T) = -1/2, we can solve for cos(T):
(-1/2)^2 + cos^2(T) = 1
1/4 + cos^2(T) = 1
cos^2(T) = 1 - 1/4
cos^2(T) = 3/4

Since T is in quadrant IV, cos(T) must be positive:
cos(T) = √(3/4) = √3/2

3. Finally, calculate the value of tan(S + T):
Using the sum identity for tangent: tan(S + T) = (tan(S) + tan(T)) / (1 - tan(S) * tan(T))

Since we have the values of sin(s), cos(s), and cos(T), we can find the values of tan(S) and tan(T):
tan(S) = sin(s) / cos(s) = (√8/3) / (1/3) = √8
tan(T) = sin(T) / cos(T) = (-1/2) / (√3/2) = -1/√3 = -√3/3

Now we can substitute these values into the tangent sum identity formula:
tan(S + T) = (tan(S) + tan(T)) / (1 - tan(S) * tan(T))
tan(S + T) = (√8 + (-√3/3)) / (1 - (√8)*(-√3/3))
tan(S + T) = (√8 - √3/3) / (1 + √8/3)

This is the exact value of tan(S + T) using the provided information.