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. *

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*Related*

Try this:

sin^6 – sin^4 +sin^4.cos^2

sin^6 – sin^4.(1 – cos^2)

sin^6 – sin^4.(sin^2)

sin^6 – sin^6 = 0

Three steps

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Nice. A more concise solution!

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The notation can really get in the way with these things. I wanted to write it as s^6 – s^4 +s^4.c^4 with the exponents in the proper places. Makes life much easier.

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Especially when written without the typos !

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My first thought was to factor out the monomial:

sin^4 (sin^2 + cos^2 -1)

then the 0 jumped out at me.

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