- #1

relskid

- 33

- 0

here's the proof:

[tex]\tan(x-y) + \tan(y-z) + \tan(z-x) = \tan(x-y)\tan(y-z)\tan(z-x)[/tex]

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 [tex]sin^2x + cos^2x = 1[/tex] and the like.

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

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

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:

[tex]\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)}[/tex]

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.

[tex]\tan(x-y) + \tan(y-z) + \tan(z-x) = \tan(x-y)\tan(y-z)\tan(z-x)[/tex]

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 [tex]sin^2x + cos^2x = 1[/tex] and the like.

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

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

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:

[tex]\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)}[/tex]

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.

Last edited: