Catalyzes the desaturation of oleic acid (Delta(9)-18:1) to linoleic acid (Delta(9), Delta(12)-18:2). (9Z)-octadecenoyl-CoA + 2 Fe(II)-[cytochrome b5] + 2 H(+) + O2 = (9Z,12Z)-octadecadienoyl-CoA + 2 Fe(III)-[cytochrome b5] + 2 H2O (9Z)-hexadecenoyl-CoA + 2 Fe(II)-[cytochrome b5] + 2 H(+) + O2 = (9Z,12Z)-hexadecadienoyl-CoA + 2 Fe(III)-[cytochrome b5] + 2 H2O Lipid metabolism; polyunsaturated fatty acid biosynthesis. The histidine box domains may contain the active site and/or be involved in metal ion binding. Belongs to the fatty acid desaturase type 1 family.