Question as interpreted:
Prove that sin^(2/3)(a)+cos^(2/3)(a)=(2/m)^(2/3)
given cos(b)=(1-m^2)/3 and tan^3(b/2)=tan(a).
Check on premises: we need to be sure that the correct interpretation of the question has been assumed. To do this, let's put m=1, then cos(b)=0 and b=π/2 is a solution. Tan^3(b/2)=1=tan(a) so a=π/4.
sin^(2/3)(a)+cos^(2/3)(a)=2^(2/3); sin(a)=cos(a)=√2/2, and the left side evaluates as 2^(2/3). Therefore the interpretation is confirmed for m=1.
Now put m=2, so b=π. Unfortunately this makes b/2=π/2, and tan(a) and tan(b/2) are infinite. However, this still satisfies the equation (1=1).
Now let b=7π/16, cos(b)=0.1951 approx and m=√(1-3cos(b))=0.6440 approx; tan(a)=0.5527 approx, and a=0.5049 radians approx.
The left side of the equation evaluates as 1.5312 and the right side as 2.1286. So the equation is not satisfied. The interpretation therefore has to be rejected.
To continue with a solution, the assumption has to be revised, which may be an impractical approach. The questioner needs to check and clarify the given conditions, then resubmit the question.
