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Principle Of Mathematical Induction

Question

Show that: $cos open parentheses fraction numerator 3 straight pi over denominator 2 end fraction plus straight x close parentheses space cos left parenthesis 2 straight pi plus straight x right parenthesis space open square brackets cot open parentheses fraction numerator 3 straight pi over denominator 2 end fraction minus straight x close parentheses plus cot left parenthesis 2 straight pi plus straight x right parenthesis close square brackets space equals space 1$

Solution

cos open parentheses fraction numerator 3 straight pi over denominator 2 end fraction plus straight x close parentheses equals sinx

WiredFaculty

cos left parenthesis 2 straight pi plus straight x right parenthesis space equals space cos left parenthesis 2 straight pi plus straight x minus 2 straight pi right parenthesis

= cosx (∵ cos x is periodic)

cot open parentheses fraction numerator 3 straight pi over denominator 2 end fraction minus straight x close parentheses space equals space tanx

cot left parenthesis 2 straight pi plus straight x right parenthesis space equals space cot left parenthesis 2 straight pi plus straight x minus 2 straight pi right parenthesis space equals space cotx

∴ L.H.S. =

space space space space space space sinx. space cosx. space left square bracket tanx plus cotx right square bracket space equals space sinx space cosx open square brackets sinx over cosx plus cosx over sinx close square brackets

sinx space cosx open square brackets fraction numerator sin squared straight x plus cos squared straight x over denominator sinx space cosx end fraction close square brackets equals 1 space equals space space straight R. straight H. straight S.