@Carl Love 

...constructing all pairs from the set of indices and omitting irrelevant ones is nothing new to me. But as a beginner, I'm still a long way from implementing this algorithm as professionally as You do. So, there's still a lot to learn :-) .

@dharr 

I can do the rest on my own :-) .

@dharr @janhardo 

I'll take a closer look at this to learn some more Maple techniques. There are some commands in there that I'm still unfamiliar with. So, thank You very much!

@janhardo 

......means, according to the formula in your graph, whether the left side is equal to the right side.

@janhardo 

...even at my age, it's still easy to do:
Multiply both sides of the statement by (2*cos(2^n*x-1)),
apply the third binomial formula to the numerator on the left side,
transform 4*cos^2(2^n*x)-1=2*(cos^2(2^n*x)-1)+1,
apply the half-angle theorem to the parentheses on the right side in the previous line, and I'm done.
My question remains:
Can Maple verify the original statement?

@Carl Love 

...after my mistake.

On
https://www.youtube.com/watch?v=WevMIH6OoQc
I found an exotic-looking addition theorem. I couldn't quickly find any errors in the proof presented there. Therefore, for practice purposes, I wanted to try to reproduce it using symbolic calculations in the Maple-world. Then should be, for the terms "term1" and "term2" in the variable x, term1 = term2. And that's why I don't understand the "false" statement. I'll probably have to use a magnifying glass and tweezers.

Thank you for your advice.

Things get puzzling for x=t/2^n. In the attached file, (3) yields the desired result, but (4) and (5) are contradictory.test2.mw
 

Download test2.mw

@mmcdara 

Thanks for both answers.
BTW
This also proved a well-known addition theorem for the angle sum in the cosine :-).

@vv 

Out of habit, I also tried "derive" with this problem – unsuccessfully. Therefore, in this collection of challenging problems, the tedious solution must be taken "by foot."
The solution for the limit is (2*cos(t)+1)/3.

@acer @dharr

...calculating in the same continuously open worksheet is made again and again ..., the simple result term (6*n-5)/16 no longer occurs; the calculation stops before it. What does this mean?

@dharr 

I would never have thought of the last two simplify commands, thank you very much. The variety of commands created by nesting is impressive.

@vv 

Please show your solution, I would like to try to learn it. Interestingly, simple result terms arise for all even exponents.

@janhardo 

BTW:

One of the typical problems is, for example, the following:
Determine the limit of
cos(pi*sqrt(x^2+x+1)) for x-->infinity.
Can Maple calculate this directly? On the other hand, a verifiable, step-by-step solution is required, one that doesn't deliver the answer immediately at the push of a button. One possible solution ultimately contains the answer to the original question.

Using

"assume(k, integer);
eval(sin(k*Pi));
                               0"

@nm 

It is an ODE with separated variables - almost a mental arithmetic task if you know some antiderivatives.