Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Show that:

cos straight pi over 7. space cos fraction numerator 2 straight pi over denominator 7 end fraction. space cos fraction numerator 4 straight pi over denominator 7 end fraction space equals space minus 1 over 8

Answer

L.H.S. = cos straight pi over 7. cos fraction numerator 2 straight pi over denominator 7 end fraction. cos fraction numerator 4 straight pi over denominator 7 end fraction space equals space fraction numerator 1 over denominator 2 sin begin display style straight pi over 7 end style end fraction open parentheses 2 sin straight pi over 7 cos straight pi over 7 close parentheses. cos fraction numerator 2 straight pi over denominator 7 end fraction cos fraction numerator 4 straight pi over denominator 7 end fraction
         equals space fraction numerator 1 over denominator 2 sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times sin open parentheses fraction numerator 2 straight pi over denominator 7 end fraction close parentheses space cos open parentheses fraction numerator 2 straight pi over denominator 7 end fraction close parentheses space cos open parentheses fraction numerator 4 straight pi over denominator 7 end fraction close parentheses    (∵ 2 space sinθ space space cosθ space equals space sin space 2 straight theta)
         equals fraction numerator 1 over denominator 4 space sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times open square brackets 2 space sin open parentheses fraction numerator 2 straight pi over denominator 7 end fraction close parentheses space cos fraction numerator 2 straight pi over denominator 7 end fraction close square brackets space cos fraction numerator 4 straight pi over denominator 7 end fraction
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           equals fraction numerator 1 over denominator 8 sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times sin open parentheses straight pi plus straight pi over 7 close parentheses space equals space fraction numerator 1 over denominator 8 sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times space minus sin open parentheses straight pi over 7 close parentheses  [∵ sin left parenthesis straight pi plus straight theta right parenthesis space equals space minus sinθ]
            space space equals 1 over 8 cross times negative 1 space equals space minus 1 over 8 space equals space straight R. straight H. straight S.
Hence,     L.H.S. = R.H.S.
         

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