Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Show that:

fraction numerator sin 11 straight x space sinx space plus space sin 7 straight x space sin 3 straight x over denominator cos 11 straight x space sinx space plus space cos space 7 straight x space sin 3 straight x end fraction = tan 8x

Answer

L.H.S. = fraction numerator sin 11 straight x space sinx space plus space sin 7 straight x space sin 3 straight x over denominator cos 11 straight x space sinx space plus space cos space 7 straight x space sin 3 straight x end fraction
      space space space space equals space minus open square brackets fraction numerator negative 2 sin 11 straight x space sinx space minus 2 sin 7 straight x space sin 3 straight x space over denominator 2 cos 11 straight x space sinx plus 2 cos space 7 straight x space sin 3 straight x end fraction close square brackets
          equals negative open square brackets fraction numerator open curly brackets cos left parenthesis 11 straight x plus straight x right parenthesis minus cos left parenthesis 11 straight x minus straight x right parenthesis close curly brackets plus open curly brackets cos left parenthesis 7 straight x plus 3 straight x right parenthesis minus cos left parenthesis 7 straight x minus 3 straight x right parenthesis close curly brackets over denominator open curly brackets sin left parenthesis 11 straight x plus straight x right parenthesis minus sin left parenthesis 11 straight x minus straight x right parenthesis close curly brackets plus open curly brackets sin left parenthesis 7 straight x plus 3 straight x right parenthesis minus sin left parenthesis 7 straight x minus 3 straight x right parenthesis close curly brackets end fraction close square brackets
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           equals negative open square brackets fraction numerator cos 12 straight x space minus space cos 4 straight x over denominator sin 12 straight x space minus space sin 4 straight x end fraction close square brackets space equals space minus open square brackets fraction numerator negative 2 space sin open parentheses begin display style fraction numerator 12 straight x plus 4 straight x over denominator 2 end fraction end style close parentheses sin open parentheses begin display style fraction numerator 12 straight x minus 4 straight x over denominator 2 end fraction end style close parentheses over denominator 2 space cos open parentheses begin display style fraction numerator 12 straight x plus 4 straight x over denominator 2 end fraction end style close parentheses sin open parentheses begin display style fraction numerator 12 straight x minus 4 straight x over denominator 2 end fraction end style close parentheses end fraction close square brackets
             equals fraction numerator sin 8 straight x space sin 4 straight x over denominator cos 8 straight x space sin 4 straight x end fraction space equals space fraction numerator sin 8 straight x over denominator cos 8 straight x end fraction space equals space tan 8 straight x space equals space straight R. straight H. straight S.
Hence,       L.H.S. = R.H.S.

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