If `sin(sinx+cosx)=cos(cosx-sinx),` and largest possible value of `sinx is pi/k ,` then the value of `k` is______.

Answer

4

Explanation

`sin (sin x+ cos x)= cos (cos x- sin x)`
or `cos (cos x - sin x)= cos (pi/2 - (sin x+ cos x))`
or `cos x - sin x= 2n pi pm (pi/2 - sin x - cos x)`
Taking +ve sign, we get
`cos x- sin x=2n pi +pi/2- sin x- cos x`
`rArr cos x= n pi +pi/4`
For `n=0, cos x=pi/4`, which is the only possible value
`:. sin x= ((sqrt(16-pi^(2))))/4` ...(i)
Taking -ve sign, we get
`sin x = pi/4` ...(ii)
From (i) and (ii), we get `pi/4` as the largest value.
Hence, `k=4`.