Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Show that:  left parenthesis sin space 3 straight x space plus space sin space straight x right parenthesis space sin space straight x space plus space left parenthesis cos space 3 straight x space minus space cos space straight x right parenthesis space cosx space equals space 0

Answer

L.H.S. = left parenthesis sin space 3 straight x space plus space sin space straight x right parenthesis space sin space straight x space plus space left parenthesis cos space 3 straight x space minus space cos space straight x right parenthesis space cos space straight x
          = open square brackets 2 space sin open parentheses fraction numerator 3 straight x plus straight x over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator 3 straight x minus straight x over denominator 2 end fraction close parentheses close square brackets sinx space plus space open square brackets negative 2 sin open parentheses fraction numerator 3 straight x plus straight x over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator 3 straight x minus straight x over denominator 2 end fraction close parentheses close square brackets cosx
     space space space space space space equals space 2 space sin 2 straight x space cosx space sinx space minus space 2 space sin space 2 straight x space sinx space cosx space equals space 0 space equals space straight R. straight H. straight S.
Hence, L.H.S. = R.H.S.

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