# Crazy Tangent Proof: Solving for $\tan(x-y) + \tan(y-z) + \tan(z-x)$

• relskid
In summary, the author worked out that sin^2x + cos^2x = 1, and then converted it to sin and cos. He was stuck after getting all the terms in sin and cos. After doing a change of variable, he was able to solve the equation.
relskid
here's the proof:

$$\tan(x-y) + \tan(y-z) + \tan(z-x) = \tan(x-y)\tan(y-z)\tan(z-x)$$

in the chapter, we learned the sin, cos, tan addition laws, so I'm assuming that we're basically limited to use those and the fundamental trig functions such as $$sin^2x + cos^2x = 1$$ and the like.

i decided to work on the right side of the equation. i first converted the tan into sin/cos:

$$[\frac{\sin(x-y)}{\cos(x-y)}][\frac{\sin(y-z)}{\cos(y-z)}][\frac{\sin(z-x)}{\cos(z-x)}]$$

i worked out the numerator (the denominator is a real pain), and as you can imagine, i got a bunch of different sines and cosines. anywho, after stuff canceled out and the like, i was left with:

$$\frac{-\sinx\sin^2z\cosx\cos^2y - \sin^2y\sinz\cos^2x\cosy + \siny\sin^2z\cos^2x\cosy - \sin^2x\siny\cosy\cos^2z + \sin^2x\sinz\cos^2y\cosz + \sinx\sin^2y\cosx\cosy\cosz}{\cos(x-y)\cos(y-z)\cos(z-x)}$$

right now, I'm stuck. if anyone can help, i'd be very grateful. :)

ps if the latex is not working, then i will just switch to the good old fashion messy look.

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Use the identity $$\tan(x-y) = \frac{\tan x - \tan y}{1+\tan x \tan y}$$ and work from the left side.

Last edited:
If you know the identities for $$\sin\left(x\pm y\right)$$ and $$\cos\left(x\pm y\right)$$, why not convert the LHS to sin and cos and see what it simplifies to.

Edit: Convert to sin and cos AFTER you use the identity provided by the courtigrad.

ok...

edit: I'm not getting the coding right, so I'm just going to write it all out.

[(sin^2x/cos^2x)(sinz/cosz) - (siny/cosy) - (sin^2y/cos^2y)(sinz/cosz) - (sinx/cosx)(sin^2y/cos^2y)(sinz/cosz) + (sinx/cosx)(sin^2y/cos^2y) - (sinx/cosx)(sin^2z/cos^2z) + (siny/cosy)(sin^2z/cos^2z) + (sinx/cosx)(sin^2y/cos^2y)(sin^2z/cos^2z) - (sin^2x/cos^2x)(siny/cosy)]

all divided by

[1 + (siny/cosy)(sinz/cosz) + (sinx/cosx)(siny/cosy) + (sinx/cosx)(sin^2y/cos^2y)(sinz/cosz) + (sinx/cosx)(siny/cosy)(sin^2z/cos^2z) + (sin^2x/cos^2x)(siny/cosy)(sinz/cosz) + (sin^2x/cos^2x)(sin^2y/cos^2y)(sin^2z/cos^2z)]

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You said that you know the tan addition law:
$$tan(a+ b)= \frac{tan(a)+ tan(b)}{1- tan(a)tan(b)}$$
so use that- don't go back to sine and cosine!

Of course, replacing b by -b, and remembering that tan(-b)= -tan(b),
$$tan(a-b)= \frac{tan(a)- tan(b)}{1+ tan(a)tan(b)}$$
as Coutrigrad said. Use that three times.

i was just doing what neutrino suggested. :P

i have it all still in tangent written on paper, but I'm still not seeing anything that can be used.

anywho... converted back to tangent. :(

$$\frac{(\tan^2 x \tan z) - (\tan y) - (\tan^2 y \tan z) - (\tan x \tan^2 y \tan z) + (\tan x \tan^2 y) - (\tan x \tan^2 z) + (\tan y \tan^2 z) + (\tan x \tan^2 y \tan^2 z) - (\tan^2 x \tan y)}{1 + (\tan y \tan z) + (\tan x \tan y) + (\tan x \tan^2 y \tan z) + (\tan x \tan y \tan^2 z) + (\tan^2 x \tan y \tan z) + (\tan^2 x \tan^2 y \tan^2 z)}$$

edit: can anyone explain to me why some of the terms are not showing up?

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Now do the same thing for the right hand side, and you can check to see how you want to re-arrange the terms you got out of the left hand side to show it equals the right hand side

I highly recommend doing a change of variable.

can anyone explain to me why some of the terms are not showing up?
Because you've issued the (nonexistant) command

\tany

\tan

followed by a y.

(In other words, you need to use spaces)

## 1. What is a "Crazy Tangent Proof"?

A Crazy Tangent Proof is a mathematical proof that involves manipulating and solving for various tangent functions in an equation. It can involve complex algebraic techniques and is often used to solve for unknown variables in trigonometric equations.

## 2. How do you solve for $\tan(x-y) + \tan(y-z) + \tan(z-x)$?

To solve for this expression, you must use the tangent addition and subtraction formulas to simplify each individual tangent term. Then, you can combine like terms and use algebraic techniques to solve for the unknown variables.

## 3. What is the purpose of solving for $\tan(x-y) + \tan(y-z) + \tan(z-x)$?

Solving for this expression can help in solving various trigonometric equations and can also be used to prove other mathematical theorems. It is a useful tool for understanding the relationships between tangent functions and angles.

## 4. Are there any special cases or restrictions to consider when solving for $\tan(x-y) + \tan(y-z) + \tan(z-x)$?

Yes, there are certain restrictions to consider when solving for this expression. First, the angles involved must be within the domain of tangent functions, which is between -π/2 and π/2. Additionally, if any of the angles involved are equal to π/2 or -π/2, the tangent function will be undefined.

## 5. Are there any tips or tricks for solving "Crazy Tangent Proofs"?

One tip is to always start by simplifying each individual tangent term using the addition and subtraction formulas. It can also be helpful to rewrite the expression in terms of sine and cosine functions, as these are often easier to manipulate. Additionally, it is important to double check your work and make sure all restrictions are considered.

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