(1+cosπ/8)(1+cos3π/8)(1+cos5π/8)(1+cos7π/8)=1/8

 

Asked by rajkumarpandey3783 | 6th Jul, 2018, 09:02: PM

Expert Answer:

begin mathsize 16px style open parentheses 1 plus cos straight pi over 8 close parentheses open parentheses 1 plus cos fraction numerator 3 straight pi over denominator 8 end fraction close parentheses open parentheses 1 plus cos fraction numerator 5 straight pi over denominator 8 end fraction close parentheses open parentheses 1 plus cos fraction numerator 7 straight pi over denominator 8 end fraction close parentheses
equals open parentheses 1 plus cos straight pi over 8 close parentheses open parentheses 1 plus cos fraction numerator 3 straight pi over denominator 8 end fraction close parentheses open parentheses 1 plus cos open parentheses straight pi minus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close parentheses open parentheses 1 plus cos open parentheses straight pi minus straight pi over 8 close parentheses close parentheses
equals open parentheses 1 plus cos straight pi over 8 close parentheses open parentheses 1 plus cos fraction numerator 3 straight pi over denominator 8 end fraction close parentheses open parentheses 1 minus cos fraction numerator 3 straight pi over denominator 8 end fraction close parentheses open parentheses 1 minus cos straight pi over 8 close parentheses
equals open parentheses 1 plus cos straight pi over 8 close parentheses open parentheses 1 minus cos straight pi over 8 close parentheses open parentheses 1 plus cos fraction numerator 3 straight pi over denominator 8 end fraction close parentheses open parentheses 1 minus cos fraction numerator 3 straight pi over denominator 8 end fraction close parentheses
equals open parentheses 1 minus cos squared straight pi over 8 close parentheses open parentheses 1 minus cos squared fraction numerator 3 straight pi over denominator 8 end fraction close parentheses
equals open parentheses 1 minus cos squared straight pi over 8 close parentheses open parentheses 1 minus cos squared open parentheses straight pi over 2 minus straight pi over 8 close parentheses close parentheses
equals open parentheses 1 minus cos squared straight pi over 8 close parentheses open parentheses 1 minus sin squared straight pi over 8 close parentheses
equals 1 minus sin squared straight pi over 8 minus cos squared straight pi over 8 plus sin squared straight pi over 8 cos squared straight pi over 8
equals 1 minus open parentheses sin squared straight pi over 8 plus cos squared straight pi over 8 close parentheses plus sin squared straight pi over 8 cos squared straight pi over 8
equals 1 minus 1 plus sin squared straight pi over 8 cos squared straight pi over 8
equals sin squared straight pi over 8 cos squared straight pi over 8
equals 1 fourth cross times open parentheses 2 sin straight pi over 8 cos straight pi over 8 close parentheses squared
equals fraction numerator sin squared straight pi over 4 over denominator 4 end fraction
equals 1 fourth cross times 1 half
equals 1 over 8 end style

Answered by Sneha shidid | 11th Jul, 2018, 10:05: AM