Hints: Competing risks model

In (a) you can express the desired probability as an integral of the joint density of \((T_1,T_2)\) over the appropriate region in the \((T_1,T_2)\)-plane. Keep in mind that \(T_1\) and \(T_2\) are independent. In (b) you have multiple options, but can for example start with the limit-definition of the hazard rate (where you'll have the probability \(P(t \leq T < \Delta t, D = j | T \geq t)\) for the cause-specific hazard rate).