arcsin(−35)=x means sinx=(−35)=−0.6.
As sinx=0.6 for x=36.87o and sine is negative in third and fourth quadrant, x=180⊕36.87o or 216.87o and x=360o−36.87o=323.13o.
arctan(512)=x means tanx=(512).
As tanx=512 for x=22.62o and tan is positive in first and third quadrant, x=22.62o or x=180o+22.62o.or 202.62o.
Hence cos{2arcsin(−35)−arctan(512)]=cos[2×216.87o−22.62o]=cos411.12o=cos51.12o=0.6277 or
cos{2arcsin(−35)−arctan(512)]=cos[2×216.87o−202.62o]=cos411.12o=cos231.12o=−0.6277 or
cos{2arcsin(−35)−arctan(512)]=cos[2×323.13o−22.62o]=cos623.64o=−0.1108 or
cos{2arcsin(−35)−arctan(512)]=cos[2×323.13o−202.62o]=cos443.64o=0.1108