Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove the following:

space space space cot space 4 straight x space left parenthesis sin space 5 straight x space plus space sin space 3 straight x right parenthesis space equals space cot space straight x space left parenthesis sin space 5 straight x space minus space sin space 3 straight x right parenthesis


Answer

L.H.S. = space space space cot space 4 straight x space left parenthesis sin space 5 straight x space plus space sin space 3 straight x right parenthesis space equals space cot space straight x space left parenthesis sin space 5 straight x space minus space sin space 3 straight x right parenthesis 
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          = fraction numerator cos space 4 straight x over denominator sin space 4 straight x end fraction left square bracket 2 space sin 4 straight x space cosx right square bracket space equals space 2 space cos 4 straight x space cosx                           ...(i)
R.H.S. = space space space cot space straight x space left parenthesis sin space 5 straight x minus space sin space 3 straight x right parenthesis space equals space cosx over sinx open square brackets 2 space cos open parentheses fraction numerator 5 straight x plus 3 straight x over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator 5 straight x minus 3 straight x over denominator 2 end fraction close parentheses close square brackets
          =  space space cosx over sinx space left square bracket 2 space cos 4 straight x space sinx right square bracket space equals space 2 space cos 4 straight x space cosx                           ...(ii)
From (i) and (ii), we have
                L.H.S. = R.HS.

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