#sin^-1x# means an angle whose sine ratio is #x#. If angle is #A#, then it means #sinA=x#.
Further, although there may be number of values of #A#, for whom sine is #x# - as all trigonometric ratios have a cycle of #2pi# radians, the range for inverse ratios is limited. While for sine, tangent, cosecant and cotangent range is #[-pi/2.pi/2]#, range for cosine and secant ratios, it is #[0,pi]#.
As #sin(pi/3)=sqrt3/2#, we have #sin^-1(sqrt3/2)=pi/3# or #60^o#
and #cos(sin^-1(sqrt3/2))=cos(pi/3)=1/2#
Note: It does not matter, whether we write angle in radians or degrees as irrespective of unit used, cosine is a ratio and
even #cos(sin^-1(sqrt3/2))=cos60^o=1/2#
