Thread:HMcCoy/@comment-25501845-20151025131611/@comment-24808864-20151026001420

This one is not terrible:

First, if you are given that $$\csc{\theta}-\cot{\theta}=\sqrt{2}\cot{x}$$, you can substitute sines and cosines for these trig functions and isolate $$\cos{\theta}$$. If you do this correctly and simplify completely you will find that $$\cos{\theta}=\sqrt{2}-1$$.

Take this fact, add one to both sides to yield $$1+\cos{\theta}=\sqrt{2}$$, then divide throughout by $$\sin{\theta}$$. The result you wish follows immediately.