Let y=arcsin(x-1), 0<=x<=2 (where <= means less than or equal to). Find x in terms of y, and show that dx/dy=cos(y).

First, the question asks us to find x in terms of y, or in other words, rearrange the equation to be in the form 'x=', rather than 'y=', as it is currently. We start with y=acrsin(x-1), as given in the question. We don't want any operations being done on the x, such as addition, multiplication, or sine/cosine, so first we apply the 'sine' operation to both sides: sin(y)=sin(arcsin(x-1)). But sin and arcsin are inverse operations, so they cancel out on the right hand side. Therefore, we have: sin(y)=x-1. Here, we add 1 to both sides, and end up with: sin(y)+1=x. We have found x in terms of y, as our equation is of the form 'x='. The latter part of the question asks us to find dx/dy, and show that it is equal to cos(y). This just means we want to find the derivative of x, with respect to y, so we will differentiate our previous answer, x=sin(y)+1. We have x=sin(y)+1, and we want to differentiate with respect to y. The derivative of the left hand side, x, with respect to y, is dx/dy. For the right hand side we have that the derivative of sin(y), with respect to y, is cos(y). The derivative of a constant, in this case '+1' is 0. We conclude that: dx/dy=cos(y). We have shown the answer given in the question.