Question

If $x+\sin y=2020$ and $x+2020\cos y=2019$, where $0\le y\le \frac{\pi }{2},$ then the value of $[x+y]$ is
( $[.]$ denotes the greatest integer function )

Answer 1

$x+\sin y=2020\cdots \left(1\right)x+2020\cos y=2019\cdots \left(2\right)$
From eqns. $\left(1\right)$ and $\left(2\right),$
$\sin y-2020\cos y=1$
$\Rightarrow \sin y=1+2020\cos y$

As $0\le y\le \frac{\pi }{2}$ and we know that
$\sin y\le 1\text{and}1+2020\cos y\ge 1$
So, they will be equal only when,
$\sin y=1\text{and}\cos y=0$
$\Rightarrow y=\frac{\pi }{2}$

Using equation $\left(1\right)$,
$x=2020-\sin y=2019$
$[x+y]=[2019+\frac{\pi }{2}]=2019+\left[\frac{\pi }{2}\right]=2019+1=2020$