You need not solve for beta and you can also get the answer.
\(\sin \beta - \cos^2 \beta = 0\\ \sin \beta = \cos^2 \beta --- (1) \\\sin^2 \beta = \cos^4 \beta ---(2) \\\sin \beta - (1 - \sin^2 \beta) = 0\\ \therefore \sin \beta + \sin^2\beta = 1 ---(3) \\\text{Substitute (1),(2) into (3)} \\\cos^4 \beta +\cos^2 \beta = 1\\\)
Then what does \((\cos^4\beta+\cos^2\beta)^3 =\)?
Use the identity \((a+b)^3 = a^3 + 3a^2b+3ab^2+b^3\)
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