what are the six trig double identities if tanx=2 and cosx<0

Question

I think i got it right, but i want to make sure, any answers would be greatly appreciated

Answer 1

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.