Question

Find the numbers of pairs of interger $(x,y)$ that satisfy the following two equations:
$\{\begin{array}{c}\cos \left(xy\right)=x\\ \tan \left(xy\right)=y\end{array}$ is

• A. 1
• B. 2
• C. 4
• D. 6

Answer 1

The correct option is A 1

$cos(x,y)=x$
And
$tan(x,y)$
$=\frac{sin(x,y)}{cos(x,y)}$
$=\frac{sin(x,y)}{x}$
$=y$
Or
$sin(x,y)=xy$.
Now
$si{n}^2(x,y)+co{s}^2(x,y)=1$
Or
$x^2+x^2{y}^2-1=0$
Or
$x^2(1+y^2)-1=0$
Now
$-1\le x\le 1$
Or
$0\le x^2\le 1$
Or
$0\le \frac{1}{1+y^2}\le 1$
Or
$1\le 1+y^2$
Or
$y^2\ge 0$
And
$-1\le sin\left(x\right)\le 1$
Or
$-1\le xy\le 1$
Or
$-1\le \frac{y}{\sqrt{1+y^2}}\le 1$
Or
$-\sqrt{1+y^2}\le y$ and $y^2\le 1+y^2$
From both of the above we get $y=0$
Hence $x=1$.
Hence only 1 number.