Answer:(4 + 3√3) / 10(-3 − 4√3) / 10(48 + 25√3) / 39Step-by-step explanation:First we need to find sin α and cos α.One way is to recognize that tan α = -4/3 corresponds to a 3-4-5 triangle. Since α is in the second quadrant:sin α = 4/5cos α = -3/5Alternatively, we can use Pythagorean identities:1 + tan² α = sec² α1 + (-4/3)² = sec² αsec α = -5/3cos α = -3/5Then use definition of tangent to find sine:tan α = sin α / cos α-4/3 = sin α / (-3/5)sin α = 4/5Next, we need to use the same process to find sin β and tan β.Since cos β = 1/2 and β is in the fourth quadrant, β = 5π/3. So sin β = -√3/2, and tan β = -√3.Or, using Pythagorean identities:sin² β + cos² β = 1sin² β + (1/2)² = 1sin β = -√3/2Using definition of tangent:tan β = sin β / cos βtan β = (-√3/2) / (1/2)tan β = -√3Now we're ready to start solving using angle sum/difference formulas.4. sin(α+β)sin α cos β + sin β cos α(4/5) (1/2) + (-√3/2) (-3/5)4/10 + 3√3/10(4 + 3√3) / 105. cos(α−β)cos α cos β + sin α sin β(-3/5) (1/2) + (4/5) (-√3/2)-3/10 − 4√3/10(-3 − 4√3) / 106. tan(α+β)(tan α + tan β) / (1 − tan α tan β)(-4/3 + -√3) / (1 − (-4/3) (-√3))(-4/3 − √3) / (1 − 4√3/3)(-4 − 3√3) / (3 − 4√3)Rationalizing the denominator:(-4 − 3√3) / (3 − 4√3) × (3 + 4√3) / (3 + 4√3)(-12 − 16√3 − 9√3 − 36) / (9 − 48)(-48 − 25√3) / -39(48 + 25√3) / 39