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If $\theta+\phi+\psi=\tfrac{1}{2}\pi,$ show that $$ \sin^{2}\theta+\sin^{2}\phi+\sin^{2}\psi+2\sin\theta\sin\phi\sin\psi=1. $$
Below are some hints about how you might approach this part of the question.
By taking $\theta=\phi=\tfrac{1}{5}\pi$ in this equation, or otherwise, show that $\sin\tfrac{1}{10}\pi$ satisfies the equation $$ 8x^{3}+8x^{2}-1=0. $$ Again, here are some hints.
Based on STEP Mathematics I, 1991, Q1. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.