In the course of evaluating the antiderivative

,

I’ve accidentally stumbled on a very curious looking trigonometric identity:

if ,

if ,

if .

The extra and are important. Without them, the graphs of the left-hand side and right-hand sides are clearly different if or :

However, they match when those constants are included:

Let’s see if I can explain why this trigonometric identity occurs without resorting to the graphs.

Since assumes values between and , I know that

,

,

and so

.

However,

,

and so and must differ if is in the interval or in the interval .

I also notice that

,

,

and so

.

However, this difference can only be equal to a multiple of , and there are only three multiples of in the interval , namely , , and .

To determine the values of where this happens, I also note that , , and are increasing functions, and so must also be an increasing function. Therefore, to determine where lies in the interval ,it suffices to determine the unique value so that . Likewise, to determine where lies in the interval ,it suffices to determine the unique value so that .

In summary, I have shown so far that

if ,

if ,

if ,

where and are the unique values so that

,

.

So, to complete the proof of the trigonometric identity, I need to show that and . I will do this in tomorrow’s post.

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