Principle of Mathematical Induction

Principle of Mathematical Induction

Question

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Answer

Let                    straight A space equals space straight pi over 8 plus straight x over 2 comma space space space space space straight B equals space straight pi over 8 minus straight x over 2
L.H.S.  =  cos squared open parentheses straight pi over 8 plus straight x over 2 close parentheses minus sin squared open parentheses straight pi over 8 minus straight x over 2 close parentheses space equals space cos squared straight A minus sin squared straight B space equals space cos left parenthesis straight A plus straight B right parenthesis. space cos left parenthesis straight A minus straight B right parenthesis
                                 space space space space equals cos open square brackets open parentheses straight pi over 8 plus straight pi over 2 close parentheses plus open parentheses straight pi over 8 minus straight pi over 2 close parentheses close square brackets. space cos open square brackets open parentheses straight pi over 8 plus straight x over 2 close parentheses minus open parentheses straight pi over 8 minus straight x over 2 close parentheses close square brackets
                                     equals cos open parentheses straight pi over 4 close parentheses. cosx space equals space fraction numerator 1 over denominator square root of 2 end fraction cosx space equals space straight R. straight H. straight S.
∴                  L.H.S. = R.H.S.
Hence,    cos squared open parentheses straight pi over 8 plus straight x over 2 close parentheses minus sin squared open parentheses straight pi over 8 minus straight x over 2 close parentheses space equals space fraction numerator 1 over denominator square root of 2 end fraction cosx

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