Asked by Sukrit Saha | 19th Jul, 2016, 10:29: PM

Expert Answer:

begin mathsize 16px style straight x over straight a equals cos left parenthesis straight theta minus straight alpha right parenthesis comma space straight y over straight b equals cos left parenthesis straight theta minus straight beta right parenthesis
LHS equals open parentheses straight x over straight a close parentheses squared plus open parentheses straight y over straight b close parentheses squared minus 2 open parentheses straight x over straight a close parentheses open parentheses straight y over straight b close parentheses cos left parenthesis straight alpha minus straight beta right parenthesis
space space space space space space space space equals cos squared left parenthesis straight theta minus straight alpha right parenthesis plus cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 space cos left parenthesis straight theta minus straight alpha right parenthesis cos left parenthesis straight theta minus straight beta right parenthesis cos left parenthesis straight alpha minus straight beta right parenthesis
space space space space space space space space equals cos squared left parenthesis straight theta minus straight alpha right parenthesis plus cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 space cos left parenthesis straight theta minus straight alpha right parenthesis cos left parenthesis straight theta minus straight beta right parenthesis cos open square brackets left parenthesis straight theta minus straight beta right parenthesis minus left parenthesis straight theta minus straight alpha right parenthesis close square brackets
space space space space space space space space equals cos squared left parenthesis straight theta minus straight alpha right parenthesis plus cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 space cos left parenthesis straight theta minus straight alpha right parenthesis cos left parenthesis straight theta minus straight beta right parenthesis left square bracket cos left parenthesis straight theta minus straight beta right parenthesis cos left parenthesis straight theta minus straight alpha right parenthesis plus sin left parenthesis straight theta minus straight alpha right parenthesis sin left parenthesis straight theta minus straight beta right parenthesis right square bracket
space space space space space space space space equals cos squared left parenthesis straight theta minus straight alpha right parenthesis plus cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 space cos squared left parenthesis straight theta minus straight alpha right parenthesis cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 cos left parenthesis straight theta minus straight alpha right parenthesis sin left parenthesis straight theta minus straight alpha right parenthesis cos left parenthesis straight theta minus straight beta right parenthesis sin left parenthesis straight theta minus straight beta right parenthesis
RHS equals sin squared left parenthesis straight alpha minus straight beta right parenthesis
space space space space space space space space equals left square bracket sin left square bracket left parenthesis straight theta minus straight beta right parenthesis minus left parenthesis straight theta minus straight alpha right parenthesis right square bracket right square bracket squared
space space space space space space space space equals left square bracket sin left square bracket left parenthesis straight theta minus straight beta right parenthesis cos left parenthesis straight theta minus straight alpha right parenthesis minus cos left parenthesis straight theta minus straight beta right parenthesis sin left parenthesis straight theta minus straight alpha right parenthesis right square bracket right square bracket squared
space space space space space space space space equals sin squared left parenthesis straight theta minus straight beta right parenthesis cos squared left parenthesis straight theta minus straight alpha right parenthesis plus cos squared left parenthesis straight theta minus straight beta right parenthesis sin squared left parenthesis straight theta minus straight alpha right parenthesis minus 2 sin left parenthesis straight theta minus straight beta right parenthesis cos left parenthesis straight theta minus straight alpha right parenthesis cos left parenthesis straight theta minus straight beta right parenthesis sin left parenthesis straight theta minus straight alpha right parenthesis
space space space space space space space space equals left square bracket 1 minus cos squared left parenthesis straight theta minus straight beta right parenthesis right square bracket cos squared left parenthesis straight theta minus straight alpha right parenthesis plus left square bracket left square bracket 1 minus cos squared left parenthesis straight theta minus straight alpha right parenthesis right square bracket right square bracket cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 sin left parenthesis straight theta minus straight beta right parenthesis cos left parenthesis straight theta minus straight alpha right parenthesis cos left parenthesis straight theta minus straight beta right parenthesis sin left parenthesis straight theta minus straight alpha right parenthesis
space space space space space space space space equals cos squared left parenthesis straight theta minus straight alpha right parenthesis plus cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 cos squared left parenthesis straight theta minus straight alpha right parenthesis cos squared left parenthesis straight theta minus straight beta right parenthesis minus 2 sin left parenthesis straight theta minus straight beta right parenthesis cos left parenthesis straight theta minus straight alpha right parenthesis cos left parenthesis straight theta minus straight beta right parenthesis sin left parenthesis straight theta minus straight alpha right parenthesis
space space space space space space space space equals LHS
Hence space proved.
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Answered by Rebecca Fernandes | 22nd Jul, 2016, 09:10: AM