Troublesome reals indeed......

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For two positive reals \(\large{a}\) and \(\large{b}\) and some \(\large{\theta}\) , we have

\(\large{a\sin \theta + b\cos \theta = 1 = a \csc \theta - b \sec \theta}\), find the value of

\(\Large{a^2 + b^2 - b^{\frac{2}{3}} + b^{\frac{4}{3}}}\)

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