Engineering Materials
posted by Z32 .
Using mechanics of materials principles (ie. equations of mechanical equilibrium applied to a freebody diagram), derive the following equations.
sigma' = sigma cos^2 θ = sigma(1+cos 2θ / 2)
and tau = sigma sin θ cos θ = sigma( sin2θ / 2)

The equations given would apply to a particular case of plane stress condition where σx and σy are the principal stresses, thus the shear stress τxy is zero. Furthermore, σx equals zero, i.e. it is a case of uniaxial tensile or compressive stress.
The derivation of the general formula for the general case is presented in the classic work "Theory of Elasticity" by Timoshenko and Goodier (chapter 2 sec. 9) and is the basis of the Mohr's circle. Therefore, the same derivation will be found in many sources including webpages.
Here's one place you may look:
http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/plane_stress.cfm#transform
The Mohr's circle is a related subject that may eventually interest you, if not for the time being.