Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Show that:

open parentheses cos space straight alpha space plus space cos space straight beta close parentheses squared space plus left parenthesis sin space straight alpha space plus space sin space straight beta right parenthesis squared space equals space 4 space cos squared space fraction numerator straight alpha minus straight beta over denominator 2 end fraction

Answer

L.H.S.  equals left parenthesis cos space straight alpha space plus space cos space straight beta right parenthesis squared plus space left parenthesis sin space straight alpha space plus space sin space straight beta right parenthesis squared
            = space space open square brackets 2 space cos open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses cos open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses close square brackets squared space plus space open square brackets 2 space sin open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses cos open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses close square brackets squared
             = 4 cos squared open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses open square brackets cos squared open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses plus sin squared open parentheses fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses close square brackets
             = 4 cos squared open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses cross times 1 space equals space 4 cos squared open parentheses fraction numerator straight alpha minus straight beta 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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