Question

Find the number of pairs of integer `(x , y)` that satisfy the following two equations: `{cos(x y)=x ` and `tan(x y)=y` (a)`1` (b)` 2` (c) `4 `(d) ` 6`

Answer 1

`cos(xy) = x and tan(xy) = y`
`=>sin(xy)/cos(xy) = y`
`=>sin(xy)/x = y`
`=>sin(xy) = xy`
Only possible value of `theta` when `sintheta = theta` is `0`.
`:. xy = 0`
`=>x = 0 or y = 0`
When `x = 0`,
`cos(0*y) = 0`
`=>cos0 = 0`, which is not possible.
`:. x = 0` is not possible.
When `y = 0`,
`cos(x*0) = x`
`=>cos0 = x`
`=>x = 1`
`:. x = 1 and y = 0` is the solution for the given equation.
`:.(1,0)` is the only pair that satisfy the given equation.