Q. Prove that cos^2(76)+cos^2(16)-cos76cos16=3/4.

Asked by Balbir | 1st Feb, 2017, 02:31: PM

Expert Answer:

begin mathsize 16px style Consider comma
LHS equals cos squared 76 plus cos squared 16 minus cos 76 space cos 16
space space space space space space equals cos squared 76 plus left parenthesis 1 minus sin squared 16 right parenthesis minus 1 half left parenthesis 2 cos 76 space cos 16 right parenthesis space.... left parenthesis straight i right parenthesis
Consider comma
space 2 cos 76 space cos 16 space equals space cos left parenthesis 76 plus 16 right parenthesis plus cos left parenthesis 76 minus 16 right parenthesis space.... left square bracket Using space cos left parenthesis straight A plus straight B right parenthesis plus cos left parenthesis straight A minus straight B right parenthesis equals 2 cosAcosB right parenthesis right square bracket
rightwards double arrow space 2 cos 76 space cos 16 space equals space cos space 92 plus cos 60
From space left parenthesis 1 right parenthesis comma space we space have
LHS equals cos squared 76 plus left parenthesis 1 minus sin squared 16 right parenthesis minus 1 half left parenthesis cos space 92 plus cos 60 right parenthesis
space space space space space space space equals left parenthesis cos squared 76 minus sin squared 16 right parenthesis plus 1 minus 1 half left parenthesis cos space 92 plus cos 60 right parenthesis space.... left parenthesis ii right parenthesis
space space space space space space space equals space cos left parenthesis 76 plus 16 right parenthesis space cos left parenthesis 76 minus 16 right parenthesis plus 1 minus 1 half left parenthesis cos space 92 plus cos 60 right parenthesis space space.... left parenthesis Since space cos squared straight A minus cos squared straight B equals cos left parenthesis straight A plus straight B right parenthesis space cos left parenthesis straight A minus straight B right parenthesis right parenthesis
space space space space space space space equals cos space 92 space cos 60 plus 1 minus 1 half left parenthesis cos space 92 plus cos 60 right parenthesis
space space space space space space space equals cos space 92 space open parentheses 1 half close parentheses plus 1 minus 1 half open parentheses cos space 92 plus 1 half close parentheses
space space space space space space space equals fraction numerator cos space 92 over denominator 2 end fraction space plus 1 minus fraction numerator cos space 92 over denominator 2 end fraction minus 1 fourth
space space space space space space space equals 1 minus 1 fourth
space space space space space space space equals 3 over 4 equals RHS
Hence space proved. end style

Answered by Rebecca Fernandes | 1st Feb, 2017, 02:47: PM