Sometimes a puzzle can look complicated, but be rather simple (see this geometry puzzle). I love puzzles like this and I particularly like to test them out on classes to try and build their problem solving ability.
Just now, I saw the following trig puzzle from brilliant.org and I love it! It’s amazing!
Have you done it yet?
How long did it take you to spot it?
My initial thought was, it’s got three terms, it’s bound to be a disguised quadratic that will factorise. A few seconds later I realised that it wasn’t. I saw the – sin^4 and suspected a difference of two squares but then a few seconds later it became clear.
If you haven’t spotted it yet, have a look at the expression rearranged:
Sin^6 + sin^4 cos^2 – sin^4
See it now? What if I rewrite it as:
Sin^4 sin^2 + sin^4 cos^2 – sin^4
I’m sure you have seen it now, but to be complete, take the common factor of the first two terms:
Sin^4 (sin^2 + cos^2) – sin^4
Obviously sin^2 + cos^2 = 1, so we’re left with:
Sin^4 – sin^4 = 0
A lovely, satisfying, simple answer to a little brain teaser. Hope you liked it as much as I did.
Cross-posted to Cavmaths here.