Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove the following:

fraction numerator sin space straight x minus sin space straight y over denominator cos space straight x plus cos space straight y end fraction space equals space tan open parentheses fraction numerator straight x minus straight y over denominator 2 end fraction close parentheses

Answer

L.H.S. = fraction numerator sin space straight x minus sin space straight y over denominator cos space straight x plus cos space straight y end fraction space equals space fraction numerator 2 cos open parentheses begin display style fraction numerator straight x plus straight y over denominator 2 end fraction end style close parentheses space sin open parentheses begin display style fraction numerator straight x minus straight y over denominator 2 end fraction end style close parentheses over denominator 2 space cos open parentheses begin display style fraction numerator straight x plus straight y over denominator 2 end fraction end style close parentheses space cos open parentheses begin display style fraction numerator straight x minus straight y over denominator 2 end fraction end style close parentheses end fraction space equals space fraction numerator sin open parentheses begin display style fraction numerator straight x minus straight y over denominator 2 end fraction end style close parentheses over denominator cos open parentheses begin display style fraction numerator straight x minus straight y over denominator 2 end fraction end style close parentheses end fraction
          space space equals space tan open parentheses fraction numerator straight x minus straight y over denominator 2 end fraction close parentheses space equals space straight R. straight H. straight S.
Hence,           L.H.S. = R.H.S.

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