## A lovely simple trigonometry puzzle

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 Betterqs here. *

Neat little identity. I saw it slightly differently by changing the given cos squared to 1 – sin squared. End result is the same of course 🙂

Aye, somebody else just said that on Facebook too. Both nice solutions. Lovely little task for y12 and 13 tomorrow I think.

Factorises immediately as sin^4 (x) [ sin^2(x) – 1 + cos^2(x) ] = 0.

Lovely concise solution as always Colin!