u can solve this question by quardadic equation or by using graph

 

 

for quardadic

 

p =cosx+cosy - cos(x+y) = (cosx+cosy) - cos(x+y)

                                     =2cos(x+y/2)cos(x-y/2) - 2cos²(x+y/2)+1

cos(x+y/2) = a

than 2acos(x-y/2)-2a²+1= p

2a^2_2acos(x-y/2)+(p-1) = 0

if the root of this equation are real than

D>=0

4cos(x-y/2)² - 8(p-1)>=0

cos(x-y/2)² >= 2(p-1)

2(p-1)<=cos(x-y/2)²<= 1

(p-1)<=1/2

p<=3/2