Explain​
Q.1. If f (x) = sin (log x), then the value of $f\left(x^2{y}^2\right)+f\left(\frac{x^2}{y^2}\right)-2f\left(x^2\right)\cos \left(log y^2\right)$ is
A) - 1
B) 0
C) 1
D) 2

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\sin \left(\mathrm{logx}\right) \\ \mathrm{f}\left({\mathrm{x}}^2{\mathrm{y}}^2\right)=\sin \left(\log \left({\mathrm{x}}^2{\mathrm{y}}^2\right)\right) \\ \mathrm{f}\left(\frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}\right)=\sin \left(\log \frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}\right) \\ \mathrm{f}\left({\mathrm{x}}^2\right)=\sin \left({\mathrm{logx}}^2\right) \\ \mathrm{Now} \\ \mathrm{f}\left({\mathrm{x}}^2{\mathrm{y}}^2\right)+\mathrm{f}\left(\frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}\right)-2\mathrm{f}\left({\mathrm{x}}^2\right)\cos \left({\mathrm{logy}}^2\right) \\ =\sin \left(\log \left({\mathrm{x}}^2{\mathrm{y}}^2\right)\right)+\sin \left(\log \frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}\right)-2\sin \left({\mathrm{logx}}^2\right)\cos \left({\mathrm{logy}}^2\right) \\ \mathrm{Now} \mathrm{this} \mathrm{will} \mathrm{hold} \mathrm{for} \mathrm{all} \mathrm{xy}\in \mathrm{Domain} \mathrm{of} \mathrm{function}, \mathrm{hence} \mathrm{put} \mathrm{x}=\mathrm{y}=1 \\ \sin \left(\log \left({\mathrm{x}}^2{\mathrm{y}}^2\right)\right)+\sin \left(\log \frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}\right)-2\sin \left({\mathrm{logx}}^2\right)\cos \left({\mathrm{logy}}^2\right) \\ =\sin \left(\log \left(1\right)\right)+\sin \left(\log 1\right)-2\sin \left(\log 1\right)\cos \left(\log 1\right) \\ \mathrm{As} \log 1=0, \mathrm{hence} \\ \sin \left(\log \left(1\right)\right)+\sin \left(\log 1\right)-2\sin \left(\log 1\right)\cos \left(\log 1\right) \\ =0+0-0 \\ =0 \end{gathered}$$

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