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[Solved] Trigonometry math question

 
Trigonometry 1 & 2
Last Post by Staff 4 years ago
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June 15, 2021 2:30 pm
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(@armor)
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June 15, 2021 2:31 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

The quantity is equal to 

\small sin^4(\frac{\Pi }{16})+sin^4(\frac{3\Pi }{16})+sin^4(\frac{\Pi }{2}-\frac{3\Pi }{16})+sin^4(\frac{\Pi }{2}-\frac{\Pi }{16})

\small =sin^4(\frac{\Pi }{16})+sin^4(\frac{3\Pi }{16})+cos^4(\frac{3\Pi }{16})+cos^4(\frac{\Pi }{16})

\small =(sin^2\frac{\Pi }{16}+cos^2\frac{\Pi }{16})^2-2sin^2\frac{\Pi }{16}cos^2\frac{\Pi }{16}+(sin^2\frac{3\Pi }{16}+cos^2\frac{3\Pi }{16})^2-2sin^2\frac{3\Pi }{16}cos^2\frac{3\Pi }{16}

\small =2-\frac{1}{2}(sin^2\frac{\Pi }{8}+sin^2\frac{3\Pi }{8})=2-\frac{1}{2}[sin^2\frac{\Pi }{8}+sin^2(\frac{\Pi }{2}-\frac{\Pi }{8})]

\small =2-\frac{1}{2}(sin^2\frac{\Pi }{8}+cos^2\frac{\Pi }{8})=\frac{3}{2}.

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