Do Principal Stresses and Strains Align in Plane Stress States?

[mm391]

Homework Statement

For a plane stress state, show that the principal stresses and strains occur in the same direction?

Homework Equations

Hookes Law for PLane Stress States
Principal Stress is tan(2θ)=2δ/(\sigmax-\sigmay)
Principal Strain is tan(2θ)=\gammaxy/(\epsilonx-\epsilony)
\gamma=\tau/G
G=E/(2(1+v))

The Attempt at a Solution

tan(2θ)*(\epsilonx-\epsilony)=tan(2θ)*(\sigmax-\sigmay)2(1+v)/2E

I cannot see how to get rid of (1+v)/E, can anyone help me out please.

 

[Chestermiller]

What do the Hooke's law equations for each of the stresses explicitly in terms of the strains reduce to when σ_z is set equal to zero?