`3^((1/2+log_3(cosx+sinx)))-2^(log_2(cosx-sinx))=sqrt(2)`

Answer

`x=2n pi+(3pi)/(4),2n pi+(pi)/(12),n in Z`

Explanation

`3^(((1)/(2)+log_(3)(cos x + sin x)))-2^(log_(2)(cos x - sin x)=sqrt(2))`
or `sqrt(3)(cos x + sin x) - (cos x - sin x) = sqrt(2)`
or `sqrt(3) cos x + sqrt(3) sin x - cos x + sin x = sqrt(2)`
or `(sqrt(3)+1)sin x + (sqrt(3)-1)cos x = (2sqrt(2))/(2)/(2)`
or `((sqrt(3)+1)/(2sqrt(2)))sin x + ((sqrt(3)-1)/(2sqrt(2)))cos x = (1)/(2)`
or `sin (x + (5pi)/(12))=(1)/(2)`
or `x + (5pi)/(12)=2n pi pm (pi)/(3), n in Z`
or `x = 2n pi pm(pi)/(3)-(5pi)/(12)`
or `x = 2n pi +(3pi)/(4), 2n pi + (pi)/(12), n in Z`