The question is definitely not trivial, but is not currently at research level because the answer has been known for a long time.   A good modern source available online <a href="http://www.mathunion.org/ICM/ICM2010.3/Main/icm2010.3.1226.1261.pdf">*here*</a> is the 2010 ICM write-up by Shrawan Kumar.   He attributes the result to the late Bert Kostant's seminal (though in retrospect overcomplicated) 1959 paper, with free access online
<a href="http://www.ams.org/mathscinet-getitem?mr=0109192">*here*</a>, deriving a closed formula for weight multiplicities in irreducible finite dimensional representations.  See Proposition (3.2) in Kumar's recent survey, where there is also a natural further refinement giving an upper bound on the multiplicity of each irreducible summand of such a tensor product.

[By the way, I included a version of this result as Exercise 24.12 in my
1972 graduate text on Lie algebras, though for some reason I can't recall now the exercise stated that it should be deduced from Steinberg's tensor product formula.  That seems misleading.]