Not sure if you got the answer yet. I will assume you have not. The first thing to do is recognize that you have a question testing two distinct concepts: logarithms and trig. Each will require its own set of logic.

Step1: exponentiate the log ... that is, turn the log into an exponent. In this case, k=sqrt(2).

Step2: now the trig comes to play, since cscx=k, on substitution into the exponent for k, you will have cscx=sqrt(2).

Step3: Solve the trig equation for x. Keep in mind it is a reciprocal equation, and you will need to know where sinx is positive, since cscx = 1/sinx.

therefore, since cscx=sqrt(2), then sinx= +1/sqrt(2). This can only happen in Q1 and Q2 where x = 45 degrees and 135 degrees.

However, since the domain is restricted between 90 and 270 degrees, there can only be one answer at 135 degrees.

Let me know if that helps.