Trigonometry - compound angles

(i)
2sin2x + cos2x = k
4sinxcosx+(1-2sin^2x)=k
4tanx+sec^2x-2tan^2x=ksec^2x
4tanx+(1+tan^2x)-2tan^2x=k(1+tan^2x)
So (1+k)tan^2 x - 4tanx + k-1 =0

(ii)
tanx1 and tanx2 are the roots of this quadratic equation in tanx

Then
tanx1+tanx2=4/(1+k)
(tanx1)(tanx2)=(k-1)/(1+k)

tan(x1 +x2)
=(tanx1+tanx2)/(1-(tanx1)(tanx2))
=[4/(1+k)]/[1-(k-1)/(1+k)]
=[4]/[(1+k)-(k-1)]
=4/2
=2

2008-03-17 10:04:47 補充：
Step的多少可以自己添加。