I'm using the Quantum Systems, Processes, and Information textbook for my quantum mechanics course and working on problem 4.4 below

I feel like there is a shortcut to this answer that I'm missing, because I've done several pages of algebra and simplifications and I can't seem to get a form that either I or Wolfram can simplify into something reasonable.

I'm calculating the uncertainty by using the equation with <A^2> - <A>^2 for some operator A.

I'm using the form of Sx below and corresponding forms of Sy and Sz

And I'm using the definition of expectation value below

However, despite double and triple checking all my algebra, this is the equation I get for the product of the two uncertainties, which is not really workable.

https://www.wolframalpha.com/input/?...cosx*sinx)%5E2))

I feel like I'm certainly missing some major shortcut.

Thanks in advance.