u can try this

LHS=[acos(x)+bsin(x)]

dividing andd multiplying with

take [a^2+b^2]^(1/2)=k

[acos(x)+bsin(x)]*k/k..(1)

when we take

a/[a^2+b^2]^(1/2)=sin(y)

we get

b/[a^2+b^2]^(1/2)=cos(y)

equation (1) than becomes

k*(sin(x+y))

here

-1=< sin(x+y) <=1

so

-k<=LHS<=k

thus proved