The complex numbers sin⁡x+icos⁡2x  and cos⁡x−isin⁡2x are conjugate to each other for

sin⁡x+icos⁡2x¯=cos⁡x−isin⁡2x⇒ sin⁡x−icos⁡2x=cos⁡x−isin⁡2x⇒ sin⁡x=cos⁡x and cos⁡2x=sin⁡2x⇒ tan⁡x=1 and tan⁡2x=1These two equations cannot hold simultaneously.