Here's an alternative approach. Note that tantheta = sin theta/costheta.
sintheta/costheta = sqrt(2)sintheta
Multiply both sides by costheta.
sintheta/costheta * costheta = sqrt(2)sintheta*costheta
Now recognize that sin2theta = 2sinthetacostheta.
sintheta = sqrt(2)sinthetacostheta
Multiply the right side by 2/2
sintheta = 2sqrt(1/2)sinthetacostheta
sintheta = sqrt(1/2)sin2theta
sintheta/(2sinthetacostheta) = 1/sqrt(2)
1/(2costheta) = 1/sqrt(2)
sqrt(2) = 2costheta
sqrt(2)/2 = costheta
This is the rationalized form of 1/sqrt(2).
1/sqrt(2) = costheta
This has solutions theta = pi/4, (7pi)/4.
Hopefully this helps!