prove that cos^2(pi/12)cos^2(pi/4)+cos^2(5pi/12)=3/2.

Asked by Balbir | 30th Jan, 2017, 09:15: PM

Expert Answer:

begin mathsize 16px style The space question space should space be space colon
Prove space that space colon space cos squared open parentheses straight pi over 12 close parentheses plus cos squared open parentheses straight pi over 4 close parentheses plus cos squared open parentheses fraction numerator 5 straight pi over denominator 12 end fraction close parentheses equals 3 over 2
LHS space equals cos squared open parentheses straight pi over 12 close parentheses plus cos squared open parentheses straight pi over 4 close parentheses plus cos squared open parentheses fraction numerator 5 straight pi over denominator 12 end fraction close parentheses
space space space space space space space space space equals space cos squared open parentheses straight pi over 4 minus straight pi over 6 close parentheses plus cos squared open parentheses straight pi over 4 close parentheses plus cos squared open parentheses straight pi over 4 plus straight pi over 6 close parentheses
space space space space space space space space space equals space open square brackets cos open parentheses straight pi over 4 close parentheses cos open parentheses straight pi over 6 close parentheses plus sin open parentheses straight pi over 4 close parentheses sin open parentheses straight pi over 6 close parentheses close square brackets squared plus open square brackets cos open parentheses straight pi over 4 close parentheses close square brackets squared plus open square brackets cos open parentheses straight pi over 4 close parentheses cos open parentheses straight pi over 6 close parentheses minus sin open parentheses straight pi over 4 close parentheses sin open parentheses straight pi over 6 close parentheses close square brackets squared
space space space space space space space space space equals space open square brackets fraction numerator 1 over denominator square root of 2 end fraction fraction numerator square root of 3 over denominator 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction 1 half close square brackets squared plus open square brackets fraction numerator 1 over denominator square root of 2 end fraction close square brackets squared plus open square brackets fraction numerator 1 over denominator square root of 2 end fraction fraction numerator square root of 3 over denominator 2 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction 1 half close square brackets squared
space space space space space space space space space equals space open square brackets fraction numerator square root of 3 over denominator 2 square root of 2 end fraction plus fraction numerator 1 over denominator 2 square root of 2 end fraction close square brackets squared plus open square brackets fraction numerator 1 over denominator square root of 2 end fraction close square brackets squared plus open square brackets fraction numerator square root of 3 over denominator 2 square root of 2 end fraction minus fraction numerator 1 over denominator 2 square root of 2 end fraction close square brackets squared
space space space space space space space space space equals space open square brackets fraction numerator square root of 3 plus 1 over denominator 2 square root of 2 end fraction close square brackets squared plus open square brackets fraction numerator 1 over denominator square root of 2 end fraction close square brackets squared plus open square brackets fraction numerator square root of 3 minus 1 over denominator 2 square root of 2 end fraction close square brackets squared
space space space space space space space space space equals space open parentheses square root of 3 plus 1 close parentheses squared over 8 plus 1 half plus open parentheses square root of 3 minus 1 close parentheses squared over 8
space space space space space space space space space equals space fraction numerator 3 plus 2 square root of 3 plus 1 plus 4 plus 3 minus 2 square root of 3 plus 1 over denominator 8 end fraction
space space space space space space space space space equals space 12 over 8
space space space space space space space space space equals space 3 over 2 equals RHS
Hence space proved.
Please space type space the space correct space question. space It space would space help space us space help space you space faster.
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Answered by Rebecca Fernandes | 31st Jan, 2017, 12:53: PM