Principle of Mathematical Induction

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Question

Prove the following:

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

Solution

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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