dear student, we will call alpha=a, beta= b, gamma= c, theta= t, phi= p.
now, observe that for proving tan^2(a/2)= tan^2(b/2)*tan^2(c/2), we simply need to eliminate t and p from the given relations. we shall make use of std formulas like
cosx= 1 – 2sin^2(x/2)= 2cos^2(x/2) – 1
and cosx= (1 – m^2)/(1+m^2) where m= tan(x/2).
given cosa= cosc*cost= cosc*(1 – 2sin^2(t/2)) 
substitute sin(t/2)= sina/2sin(p/2) (given)
so cosa= cosc*(1 – sin^2a/2sin^2(p/2)).
now, put 2sin^2(p/2)= 1 – cosp= 1 – cosa/cosb (given)
so cosa= cosc*(1 – sin^2a/(1 – cosa/cosb)).
in this relation, we have successfully eliminated t and p. now, simply write sin^2a= 1 – cos^2a and replace cosa, cosb, cosc by (1 – x^2)/(1+x^2), (1 – y^2)/(1+y^2) and (1 – z^2)/(1+z^2) where x, y, z are tan(a/2), tan(b/2), tan(c/2) respectively. hence, we have finally obtained a relation b/w tan(a/2), tan(b/2) and tan(c/2) alone. now rearrange and simplify and u ll get the reqd result, albeit i must warn you that the calculations are lengthy and prone to mistakes.
KINDLY APPROVE :))