cos(x)+sin(x)=1.4142
There are two things you have to notice here.
1) 1.4142 is sqrt(2) and cos(pi/4) = sin(pi/4) = 1/sqrt(2)
2) sin(A+B) = sinA.cosB + cosA.sinB
And you can apply both of these observations to the equation you have to solve.
Starting with cos(x)+sin(x)=1.4142
cos(x)*(1/sqrt(2)) + sin(x)*(1/sqrt(2)) = 1
cos(x).sin(pi/4) + sin(x).cos(pi/4) = 1
sin(x+pi/4) = 1 = sin(pi/2)
x+pi/4 = pi/2
x = pi/4