cos(x)<0 in quadrants 2 and 3; tan(x)>0 in quadrant 3.
Therefore we know x lies between π and 3π/2 (180º and 270°).
If tan(x)=2, then in quadrant 1, sin(x)=2/√5 or 2√5/5 and cos(x)=√5/5.
Now we need to move these into quadrant 3.
So sin(x)=-2√5/5 and cos(x)=-√5/5, tan(x)=2;
csc(x)=-√5/2, sec(x)=-√5, cot(x)=½.
If, by double-identities you mean tan(2x), etc., then:
tan(2x)=2tan(x)/(1-tan²(x))=4/(-3)=-4/3;
sin(2x)=2sin(x)cos(x)=(-4√5/5)(-√5/5)=⅘;
cos(2x)=2cos²(x)-1=⅖-1=-⅗;
cot(2x)=-¾; csc(2x)=5/4; sec(2x)=-5/3.