What is sin^2 + cos^2?

The answer to this question is 1. The reason the answer is 1 is because sin^2 + cos^2 =1 is a trig identity that makes problems in higher level calculus problems easier to solve.

Why does the principal element have no shear stress?

The principal element of a material has no shear stress as it the ideal plain inside a material. This plain is a certain angle, theta, away from the plane that is in a "planar" state of stress. Since this plane has no shear stress it allows us to draw a 3-d mohr's circle and create the entire principle element.

What is the derivative of x^2 + 2x?

This problem revolves around the power rule of derivatives. You can use the power rule for both 'x' variables in the problem. The use of the rule leads to x^2 becoming 2x. The 2x then because of the 'x' that is raised to the power of 1 just becomes the constant 2. Your final answer would then be 2x+2.