If A, B, C are angles of a triangle, then the value of e2iA    e−iC    e−iBe−iC    e2iB    e−iAe−iB    e−iA    e2iC is

Since A+B+C=π and eiπ=cos⁡π+isin⁡π=−1,ei(B+C)=ei(π−A)=−e−iA and e−i(B+C)=−eiABy taking eiA, eiB, eiC common from R1, R2 and R3, respectively,we haveΔ=−eiAe−i(A+C)e−i(A+B)e−i(B+C)eiBe−i(A+B)e−i(B+C)e−i(A+C)eiC=−eiA−eiB−eiC−eiAeiB−eiC−eiA−eiBeiCBy taking eiA, eiB, eiC common from C1, C2 and C3, respectively,we haveΔ=1−1−1−11−1−1−11=−4