`sin^(-1)((-1)/(2))+cos^(-1)((-sqrt(3))/(2))=cos^(-1)((-1)/2)`

Explanation

Let `sin^(-1)(-(1)/(2))=alpha, "where" -(pi)/(2)le alpha le (pi)/(2)`
`therefore sin alpha=(1)/(2)=-sin.(pi)/(6)`
`therefore sin alpha=sin.(-(pi)/(6))" ".......[because sin(-theta)=-sintheta]`
`therefore alpha=(pi)/(6)" "..........[because -(pi)/(2) le -(pi)/(6)le (pi)/(2)]`
`therefore sin^(-1)(-(1)/(2))=-(pi)/(6)`
Let `cos^(-1)(-(sqrt(3))/(2))=beta, "where " 0 le beta le pi`
`therefore cos beta =-(sqrt(3))/(2)=-cos.(pi)/(6)`
`therefore cos beta =cos (pi-(pi)/(6)) " "...[because cos (pi-theta)=-costheta]`
`therefore cos beta=cos.(5pi)/(6)" "......[because 0 le (5pi)/(6)le pi]`
`therefore cos^(-1) (-(sqrt(3))/(2))=(5pi)/(6)` .....(2)
Let `co^(-1)(-(1)/(2))=gamma, "where " 0 le gamma le pi`
`therefore cos gamma =-(1)/(2)=-cos.(pi)/(3)`
`therefore cos gamma =cos(pi-(pi)/(3))" "......[because cos (pi-theta)=-costheta]`
`therefore cos gamma = cos (2pi)/(3)`
`therefore gamma =(2pi)/(3)" "..........[because 0 le (2pi)/(3)le pi]`
`therefore cos^(-1)(-(1)/(2))=(2pi)/(3)` ....(3)
LHS `=sin^(-1)(-(1)/(2))+cos^(-1)(-(sqrt(3))/(2))`
`= -(pi)/(6)+(5pi)/(6)` " ".......[By(1) and (2)]
`=(4pi)/(6)=(2pi)/(3)`
`=cos^(-1)(-(1)/(2))` " ".........[By(3)]
=RHS.