If cos θ = -8/17 and θ is in quadrant iii, cos 2θ= and tan 2θ=

Note that cos theta = adj / hyp, and that we can therefore assume that adj = -8 and hyp = 17.  With theta in Q III, the opp side must be negative.  What is it?  Use the Pyth. Thm. to find the opp side: 

17^2 = (-8)^2 + opp^2    => 289 = 64 + |opp|^2.  Then, in Q III, opp = sqrt(225) = -15.

In summary, opp = -15, adj = -8 and hyp = 17.

cos 2 theta = cos theta cos theta - sin theta sin theta, or
                      (cos theta)^2 - (sin theta)^2
                   = (-8/17)^2 - (-15/17)^2 = approx 0.221 - 0.779 = -0.558 (ans)

Review:  cos 2 theta = (cos theta)^2 - (sin theta)^2
               sin 2 theta = 2(sin theta)(cos theta)

                                       2 tan theta
                tan 2 theta =
                                      1 - (tan theta)^2