How can four tangent values equal one?

[xphloem]

Homework Statement

My calculator shows tan 6. tan 42. tan 66. tan 78 =1
Ho is this possible?

Homework Equations

tan (90-x)=cot x

The Attempt at a Solution

but these angles are not supplementary. Ho is this possible?

 

[symbolipoint]

What were you expecting?

$$\[ (.1051)(.9004)(2.246)(4.7046) = 0.99993... \]$$

That seems close enough to call "1".

 

[tiny-tim]

Welcome to PF!

Hi xphloem! Welcome to PF!

How did you come across this?

If we take inverses, we can rewrite this as tan 12 tan 24 tan 48 tan 96 = 1.

So I suspect it has something to do with a regular pentagon, whose interior angles are 108º, and whose exterior angles are 72º.

But that's as far as I've got!

 

[kamerling]

It's actually 12 tan 24 tan 48 tan 96 = -1

I've been able to prove cos 12 cos 24 cos 48 cos 96 = -1/16

sin 12 sin 24 sin 48 sin 96 is equal to 1/16 but I can only prove this with a calculator

sin(24) = 2sin(12)cos(12)

sin(48) = 2sin(24)cos(24) = 4sin(12)cos(12)cos(24)

sin(96) = 2sin(48)cos(48) = 8sin(12)cos(12)cos(24)cos(48)

sin(192) = -sin(12) = 2sin(96)cos(96) = 16sin(12)cos(12)cos(24)cos(48)cos(96)

divide by 16 sin(12) to get:

cos 12 cos 24 cos 48 cos 96 = -1/16

 

[xphloem]

thanks all. I got this in a book called brain teasers. I have solved it myself. thanks for all the help!

 

[kamerling]

thanks all. I got this in a book called brain teasers. I have solved it myself. thanks for all the help!

so how did you solve it? any hints?

 

[tiny-tim]

cos 12 cos 24 cos 48 cos 96 = -1/16

ooh, kamerling, that's clever!

Your splitting the problem into cos and sin products has given me the following idea:

tanAtanBtanCtanD = -1
iff cosAcosBcosCcosD + sinAsinBsinCsinD = 0​

But (4cosAcosBcosCcosD + sinAsinBsinCsinD)

= [cos(A+B) + cos(A-B)][cos(C+D) + cos(C-D)] + [cos(A+B) - cos(A-B)][cos(C+D) - cos(C-D)]

= 2[cos(A+B)cos(C+D) + cos(A-B)cos(C-D)]

= cos(A+B+C+D) + cos(-A-B+C+D) + cos(A-B-C+D) + cos(-A+B-C+D)

(putting E = A + B + C + D)

= cosE + cos(E - 2(A+B)) + cos(E - 2(B+C)) + cos(E - 2(C+A)).

So, in particular, tanAtanBtanCtanD = -1 if A+B+C+D = 180º and:

cos2(B+C) + cos2(C+A) + cos2(A+B) = -1;​

or if A+B+C+D = 90º and:

sin2(B+C) + sin2(C+A) + sin2(A+B)) = 0.​

(For example, 12º 24º 48º and 96º, because cos72º + cos120º + cos144º = cos72º - cos60º - cos36º = -1)​

can anyone find a simpler proof
or some geometric explanation for this? ​

 

[xphloem]

so how did you solve it? any hints?

I got this from the following equations:
2 sin A sin B= cos (A-B)-cos(A+B)
2 cos A cos B=cos (A-B)+cos(A+B)

Divide the (i) by (ii)
to obtain the result :)