If tan alpha 158 and cot beta 125 for a secondquadrant ang

If tan alpha = -15/8 and cot beta = 12/5 for a second-quadrant angle alpha and a third-quadrant angle beta, find the following. Sin(alpha + beta) cos(alpha + beta) tan(alpha + beta) sin(alpha - beta) cos(alpha - beta) tan(alpha - beta)

Solution

tan alpha=-15/8 alpha is in second quadrant

sin alpha= 15/17

cos alpha= -8/17

cot beta=12/5   beta is in third quadrant

tan beta=5/12

sin beta= -5/13

cos beta= -12/13

a. sin(alpha + beta)= sin alpha cos beta+ cos alpha sin beta

                            = (15/17)(-12/13) + (-8/17)(-5/13) = -141/221

b. cos(alpha + beta)= cos alpha cos beta -sin alpha sin beta =(-8/17)(-12/13)-(15/17)(-5/13)

                                                                                       = 171/221

c.tan(alpha + beta)=sin(alpha + beta)/cos(alpha+ beta) = -141/171

d. sin(alpha-beta)= sin alpha cos beta-cos alpha sin beta = (15/17)(-12/13)-(-8/17)(-5/13)= -220/221

e. cos(alpha- beta)= cos alpha cos beta + sin alpha sin beta=(-8/17)(-12/13)+(15/17)(-5/13)= 21/221

f. tan(alpha-beta)= sin(alpha-beta)/cos(alpha-beta)= -220/21