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Question

If f(x)=cos(logx) , then show that f(1x).f(1y)12[fxy+f(xy)]=0

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Solution

f(x)cos(logx)
f(1x)=cos(log(1x)=cos(log1logx)
=cos(logx)=cos(logx)
f(1y)=cos(log(1y))=cos(log1logy)
=cos(logy)=cos(logy)
f(xy)=cos(log(xy))=cos(logxlogy)
=cos(logx)cos(logy)+sin(logx)sin(logy)
f(xy)=cos(log(xy)=cos(logx+logy)
=cos(logx)cos(logy)sin(logx)sin(logy)
Now,
f(1x).f(1y)12[f(xy)+f(xy)]
=cos(logx).cos(logy)12[2cos(logx)cos(logy)]
=cos(logx)cos(logy)cos(logx)cos(logy)
=0

1073681_1091145_ans_7a646d2b13f34a388c0c15fb36204e37.png

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