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General
Easy

Question

cos to the power of 6 space pi over 16 plus cos to the power of 6 space fraction numerator 3 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 5 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 7 pi over denominator 16 end fraction =

  1. 0
  2. 1/4
  3. 3/4
  4. 5/4

Hint:

In the question we will start simplification with the formula cos straight pi over 2 minus A equals sin A then we will further solve it using some algebraic formula and some trigonometric identities like sin squared A plus cos squared A equals 1 space a n d space 2 sin A cos A equals sin 2 A

The correct answer is: 5/4


    cos to the power of 6 space pi over 16 plus cos to the power of 6 space fraction numerator 3 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 5 pi over denominator 16 end fraction plus cos to the power of 6 space fraction numerator 7 pi over denominator 16 end fraction
equals cos to the power of 6 straight pi over 16 plus cos to the power of 6 open parentheses straight pi over 2 minus straight pi over 16 close parentheses plus cos to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 6 open parentheses straight pi over 2 minus fraction numerator 3 straight pi over denominator 16 end fraction close parentheses
equals cos to the power of 6 straight pi over 16 plus sin to the power of 6 straight pi over 16 plus cos to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction plus sin to the power of 6 fraction numerator 3 straight pi over denominator 16 end fraction space space space space space space space space space space space space space space space open square brackets cos open parentheses straight pi over 2 minus A close parentheses equals sin A close square brackets
equals open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses open parentheses sin to the power of 4 straight pi over 16 plus cos to the power of 4 straight pi over 16 minus sin squared straight pi over 16 cos squared straight pi over 16 close parentheses plus open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses open parentheses sin to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction plus cos to the power of 4 fraction numerator 3 straight pi over denominator 16 end fraction minus sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses space space space space open square brackets a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared plus b squared minus a b close parentheses close square brackets
equals open square brackets open parentheses sin squared straight pi over 16 plus cos squared straight pi over 16 close parentheses squared minus 2 sin squared straight pi over 16 cos squared straight pi over 16 minus sin squared straight pi over 16 cos squared straight pi over 16 close square brackets plus open square brackets open parentheses sin squared fraction numerator 3 straight pi over denominator 16 end fraction plus cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses squared minus 2 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction minus sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close square brackets space space space space space space space space open square brackets a squared plus b squared equals open parentheses a plus b close parentheses squared minus 2 a b close square brackets
equals open parentheses 1 minus 3 sin squared straight pi over 16 cos squared straight pi over 16 close parentheses plus open parentheses 1 minus 3 sin squared fraction numerator 3 straight pi over denominator 16 end fraction cos squared fraction numerator 3 straight pi over denominator 16 end fraction close parentheses
equals 1 minus 3 over 4 sin squared straight pi over 8 plus 1 minus 3 over 4 sin squared fraction numerator 3 straight pi over denominator 8 end fraction space space space space space space space open square brackets 4 sin squared A cos squared A equals sin squared 2 A close square brackets
equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus sin squared fraction numerator 3 straight pi over denominator 8 end fraction close square brackets
equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus sin squared open parentheses straight pi over 2 minus straight pi over 8 close parentheses close square brackets
equals 2 minus 3 over 4 open square brackets sin squared straight pi over 8 plus cos squared straight pi over 8 close square brackets space space space space space space space space space space space space space space space open square brackets sin open parentheses straight pi over 2 minus A close parentheses equals cos A close square brackets
equals 2 minus 3 over 4
equals 5 over 4

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