Question

If f(x) = cos (log x), then $f\left(x^2\right)f\left(y^2\right)-\frac{1}{2}[f\left(\frac{x^2}{y^2}\right)+f(x^2{y}^2\left)\right]$ has the value

• A. -2
• B. -1
• C. $\frac{1}{2}$
• D. None of these

Answer 1

The correct option is D None of these

$f\left(x^2\right)f\left(y^2\right)-\frac{1}{2}[f\left(\frac{x^2}{y^2}\right)+f(x^2{y}^2\left)\right]$
$=cos\left(log{x}^2\right)cos\left(log{y}^2\right)-\frac{1}{2}[cos\left(log\frac{x2}{y^2}\right)+coslog(x^2{y}^2\left)\right]$
$=cos\left(log{x}^2\right)cos\left(log{y}^2\right)-\frac{1}{2}\left[cos\right(log{x}^2-log{y}^2)+cos(log{x}^2+log{y}^2\left)\right]$
$=cos\left(log{x}^2\right)cos\left(log{y}^2\right)-\frac{1}{2}\left[2cos\right(log{x}^2\left)cos\right(log{y}^2\left)\right]=0$