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1.Relation and Function
medium
If $f(x) = \cos (\log x)$, then $f({x^2})f({y^2}) - \frac{1}{2}\left[ {f\,\left( {\frac{{{x^2}}}{2}} \right) + f\left( {\frac{{{x^2}}}{{{y^2}}}} \right)} \right]$ has the value
A
$-2$
B
$-1$
C
$1/2$
D
None of these
Solution
(d) The given expression is
$\cos \,(\log {x^2})\cos \,(\log {y^2}) – \frac{1}{2}\,\left[ {\cos \log \frac{{{x^2}}}{2} + \cos \log \frac{{{x^2}}}{{{y^2}}}} \right]$
$ = \frac{1}{2}\left[ {\cos \,(\log {x^2} + \log {y^2}) + \cos \,(\log {x^2} – \log {y^2})} \right]$
$ – \frac{1}{2}\left[ {\cos \,\log \frac{{{x^2}}}{2} + \cos \,(\log {x^2} – \log {y^2})} \right]$
$ = \frac{1}{2}\,\left[ {\cos \log {x^2}{y^2} – \cos \log \frac{{{x^2}}}{2}} \right]$.
Standard 12
Mathematics
