Principle Of Mathematical Induction

Question
CBSEENMA11014308

Prove that: cosA. space cos open parentheses straight pi over 3 minus straight A close parentheses. space cos open parentheses straight pi over 3 plus straight A close parentheses space equals space 1 fourth cos 3 straight A.

Solution

L.H.S. = cosA. space cos open parentheses straight pi over 3 minus straight A close parentheses. space cos open parentheses straight pi over 3 plus straight A close parentheses
         equals space cosA. space open square brackets cos squared open parentheses straight pi over 3 close parentheses minus sin squared straight A close square brackets
                                 open square brackets because space cos space left parenthesis straight A plus straight B right parenthesis. space cos left parenthesis straight A minus straight B right parenthesis space equals space cos squared straight A minus sin squared straight B close square brackets
          equals space cosA open square brackets open parentheses 1 half close parentheses squared minus sin squared straight A close square brackets                                    space space space space open parentheses because space cos straight pi over 3 space equals space cos 60 degree space equals space 1 half close parentheses
          equals space cosA space open square brackets 1 fourth minus sin squared straight A close square brackets space equals space cosA over 4 open square brackets 1 minus 4 sin squared straight A close square brackets
          equals space cosA over 4 open square brackets 1 minus 4 left parenthesis 1 minus cos squared straight A right parenthesis close square brackets space equals space cosA over 4 open square brackets negative 3 plus 4 space cos squared straight A close square brackets
           equals 1 fourth left square bracket 4 space cos cubed straight A space minus space 3 space cosA right square bracket space equals space 1 fourth space cos space 3 straight A space left parenthesis because space cos space 3 straight A space equals space 4 space cos cubed straight A space minus space 3 space cosA right parenthesis
Hence,  cosA. space cos open parentheses straight pi over 3 minus straight A close parentheses. space cos open parentheses straight pi over 3 plus straight A close parentheses space equals space 1 fourth cos 3 straight A
           cosA. space cos left parenthesis 60 degree minus straight A right parenthesis. space cos left parenthesis 60 degree plus straight A right parenthesis space equals space 1 fourth cos 3 straight A

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