# Product=1 homework question

• xphloem
In summary, xphloem found a way to solve the equation tan(90-x)=cot x by splitting it into cos and sin products. He found that 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. By solving for tanAtanBtanCtanD, xphloem was able to find that

## Homework Statement

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

tan (90-x)=cot x

## The Attempt at a Solution

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

What were you expecting?

$$$(.1051)(.9004)(2.246)(4.7046) = 0.99993...$$$

That seems close enough to call "1".

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!

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 :grumpy:

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

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

xphloem said:
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?

kamerling said:
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?

kamerling said:
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 :)

## 1. What is the purpose of "Product=1" in this homework question?

The purpose of "Product=1" is likely to be a placeholder value that is used in a mathematical equation, where "Product" represents the result of multiplying two or more numbers together. It could also be a variable that needs to be determined or solved for.

## 2. How do I solve this homework question with "Product=1"?

Without knowing the exact details of the homework question, it is difficult to provide a specific answer. However, to solve for "Product=1", you would need to use the given information, any relevant formulas or equations, and basic mathematical operations to determine the value of the product.

## 3. Can you provide an example of a homework question with "Product=1"?

One example of a homework question with "Product=1" could be: "Find the value of x if 3x + 2 = Product=1". In this case, "Product=1" represents the result of multiplying two numbers together, and you would need to solve for the value of x using algebraic methods.

## 4. Is "Product=1" a common notation in math homework questions?

Yes, "Product=1" is a common notation in math homework questions, particularly in algebra and calculus. It is used to represent the result of multiplication and can also be used as a placeholder value or variable in equations.

## 5. Are there any tips for solving homework questions with "Product=1"?

Some tips for solving homework questions with "Product=1" include: understanding the context of the question, identifying any given information, using relevant formulas or equations, and double-checking your calculations. It can also be helpful to break down the problem into smaller steps and to use logical reasoning to guide your approach.