Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove the following:

sin space 2 straight x space plus space 2 space sin space 4 straight x space plus space sin space 6 straight x space equals space 4 cos squared space straight x space sin space 4 straight x



Answer

L.H.S. = sin space 2 straight x space plus space 2 space sin space 4 straight x space plus space sin space 6 straight x space equals space sin space 2 straight x space plus space sin space 4 straight x space plus space sin space 4 straight x space plus space sin space 6 straight x
         = left square bracket sin space 4 straight x space plus space sin space 2 straight x right square bracket space plus space left square bracket sin space 6 straight x space plus space sin space 4 straight x right square bracket
         = 2 space sin open parentheses fraction numerator 4 straight x plus 2 straight x over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator 4 straight x minus 2 straight x over denominator 2 end fraction close parentheses plus 2 space sin open parentheses fraction numerator 6 straight x plus 4 straight x over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator 6 straight x minus 4 straight x over denominator 2 end fraction close parentheses
         = 2 space sin 3 straight x space cosx space plus space 2 space sin 5 straight x space cosx space equals space 2 space cosx space left square bracket sin space 3 straight x space plus space sin space 5 straight x right square bracket
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#6 {main}</pre>
          = 2 space cos space straight x space left square bracket 2 space sin space 4 straight x space cos space straight x right square bracket space equals space 4 space cos squared space straight x space sin space 4 straight x space equals space straight R. straight H. straight S.
Hence,    L.H.S. = R.H.S.
           

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