[tan^3(x)-cot^3(x)]/[tan(x)-cot(x)] = sec^2(x)+cot^2(x)

Yes.

Yes.

Could you show me the steps? :)

This one's a bit messy. Start by converting the tangent, cotangent, and secant functions to their sine and cosine equivalents, then simplify the resulting fractions. You'll end up with the left hand side having numerator with sine and cosine functions raised to the 6th power, but then the numerator can be divided by a term in the denominator to significantly simply things.