If . Find the value of .
Thanks.
Stage 1. What does the given information tell you? Multiply out the fractions and it becomes . Using standard trig formulas for sines and cosines , you can write this as .
Stage 2. Can you think of any examples where that condition is satisfied? One way to get is if and . That will occur for example if and . Then , and (see here) . For that example, , which simplifies (details omitted – do the calculation to check that it works) to 1.
Stage 3. Make a wild guess. Could it always be true that (given that )?
Stage 4. See if you can prove that the guess in Stage 3 is correct. If it is, then multiplying out the fractions means that we want to show that . Let's see: