Prove that:
 
cos n. cos 2 n. cos 4 n. cos 8 n space equals space fraction numerator sin 16 n over denominator 16 sin n end fraction

Asked by Olivia Adhikary | 4th Aug, 2014, 12:32: PM

Expert Answer:

cos space n. space cos space 2 n. space cos space 4 n. space cos space 8 n equals fraction numerator sin space 16 n over denominator 16 space sin space n end fraction C o n s i d e r space L H S L H S equals cos space n. space cos space 2 n. space cos space 4 n. space cos space 8 n rightwards arrow fraction numerator 2 open square brackets cos space n. space cos space 4 n close square brackets space 2 open square brackets cos space 2 n. cos space 8 n close square brackets over denominator 4 end fraction rightwards arrow A p p l y space 2 space cos space A space cos space B equals cos space open parentheses A plus B close parentheses plus cos space open parentheses A minus B close parentheses rightwards arrow fraction numerator open square brackets cos space 5 n space plus space cos space 3 n close square brackets open square brackets cos space 10 n space plus space cos space 6 n close square brackets over denominator 4 end fraction rightwards arrow fraction numerator cos space 5 n. cos space 10 n plus cos space 5 n. cos space 6 n plus space cos space 3 n. cos space 10 n plus space cos space 3 n. cos space 6 n over denominator 4 end fraction rightwards arrow fraction numerator 2 open square brackets cos space 5 n. cos space 10 n plus cos space 5 n. cos space 6 n plus space cos space 3 n. cos space 10 n plus space cos space 3 n. cos space 6 n close square brackets over denominator 8 end fraction rightwards arrow A p p l y space 2 space cos space A space cos space B equals cos space open parentheses A plus B close parentheses plus cos space open parentheses A minus B close parentheses rightwards arrow fraction numerator cos space 15 n space plus space cos space 5 n space plus thin space cos space 11 n space plus space cos space n space plus space cos space 13 n space plus space cos space 7 n space plus space cos space 9 n space plus space cos space 3 n over denominator 8 end fraction rightwards arrow fraction numerator 2 space sin space n open square brackets cos space 15 n space plus space cos space 5 n space plus thin space cos space 11 n space plus space cos space n space plus space cos space 13 n space plus space cos space 7 n space plus space cos space 9 n space plus space cos space 3 n close square brackets over denominator 16 space sin space n end fraction rightwards arrow fraction numerator 2 space cos space 15 n space sin space n plus 2 space cos space 5 n space sin space n space plus space 2 space cos space 11 n space sin space n plus plus thin space 2 space cos space n space sin space n plus space 2 space cos space 13 n space sin space n plus space 2 space cos space 7 n space sin space n space plus space 2 space cos space 9 n space sin space n plus 2 space cos space 3 n space sin space n over denominator 16 space sin space n end fraction rightwards arrow A p p l y space 2 space cos space A space sin space B equals sin space open parentheses A plus B close parentheses minus sin space open parentheses A minus B close parentheses rightwards arrow fraction numerator sin space 16 n space minus space sin space 14 n space plus thin space sin space 6 n space minus space sin space 4 n plus sin space 12 n minus sin space 10 n plus sin space 2 n space minus space 0 space plus space sin space 14 n space minus space sin space 12 n space plus thin space sin space 8 n space minus space sin space 6 n space plus space sin space 10 n space minus space sin space 8 n space plus space sin space 4 n space minus space sin space 2 n over denominator 16 space sin space n end fraction rightwards arrow A l l space l i k e space t e r m s space g e t s space c a n c e l l e d space a n d space r e m a i n i n g space t e r m space i s space sin space 16 n rightwards arrow fraction numerator sin space 16 n over denominator 16 space sin space n end fraction

Answered by Dharma Teja | 5th Aug, 2014, 09:01: AM

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