sir , please help me solving this question Question :-If f(x)=cos(logx) , then show that f(1/x)f(1/y)-1/2[f(x/y)+f(xy)]=0

Asked by Bharadwaj.R.S | 8th Aug, 2010, 12:00: AM

Expert Answer:

y = f(x) =  cos(log x)
f(1/x) = cos(log 1/x) = cos(- log x) =  cos(log x)
f(1/y) = cos(log 1/y) = cos(- log y) =  cos(log y)
f(x/y) = cos(log x/y) = cos(log x - log y) =  cos(logx)cos(logy) + sin(logx)sin(logy)
f(xy) = cos(log xy) = cos(log x + log y) =  cos(logx)cos(logy) - sin(logx)sin(logy)
Now,
f(1/x)f(1/y)  - 1/2[f(x/y) + f(xy)] =
cos(log x) cos(log y) - 1/2[cos(logx)cos(logy) + sin(logx)sin(logy) + cos(logx)cos(logy) - sin(logx)sin(logy)] =
cos(log x) cos(log y) - 1/2[cos(logx)cos(logy) + cos(logx)cos(logy)] =
cos(log x) cos(log y) - cos(logx)cos(logy) =
cos(log x) cos(log y) - cos(logx)cos(logy) =
0
Regards,
Team,
TopperLearning.
 
 
 

Answered by  | 8th Aug, 2010, 12:49: PM

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