How Do You Verify This Trigonometric Identity?

[InFiNiTeX]

cos x - cos y sin x - sin y
sin x + sin y + cos x + cos y = 0

or to see better i guess...

(cos x - cos y)/(sin x + sin y) + (sin x - siny)/(cos x + cos y) = 0

can you guys help me? I'm really stuck on this!

 

[hypermorphism]

cos x - cos y sin x - sin y
sin x + sin y + cos x + cos y = 0

or to see better i guess...

(cos x - cos y)/(sin x + sin y) + (sin x - siny)/(cos x + cos y) = 0

can you guys help me? I'm really stuck on this!

What have you tried and where are you stuck ?

 

[robphy]

Substitute $$\cos \theta = \displaystyle\frac{e^{i\theta}+e^{-i\theta}}{2}$$ and $$\sin \theta = \displaystyle\frac{e^{i\theta}-e^{-i\theta}}{2i}$$ with the understanding that $i^2=-1$.

 

[Data]

How about just combining the fractions?

 

[hypermorphism]

Substitute $$\cos \theta = \displaystyle\frac{e^{i\theta}+e^{-i\theta}}{2}$$ and $$\sin \theta = \displaystyle\frac{e^{i\theta}-e^{-i\theta}}{2i}$$ with the understanding that $i^2=-1$.

There's actually a simpler way to solve this without resorting to complex identities that the original poster may not know.

 

[robphy]

There's actually a simpler way to solve this without resorting to complex identities that the original poster may not know.

While this may be true, this exponential method never fails, of course.

 

[Integral]

In fact it is nearly trivial, obvious to casual inspection.

OP , What have you tried?

 

[InFiNiTeX]

well this is what i tried to do...

first i tried to combine them

(cos x - cos y) ( sin x - sin y)/ (sin x + sin y) (cos x + cos y)

then i got

(cos^2 x ) - (cos^2 y)/(sin x + sin y) (cos x + cos y) + (sin^2 x)(sin^2 y)/ (sin x + sin y) (cos x + cos y)

and after that part i kinda just confused myself but i just thought of this...

that part changes to

(1 - sin^2 x) - (1 - sin^2 y) / (sin x + sin y) (cos x + cos y) + (1 - cos^2 x) (1 - cos y) / (sin x + sin y) (cos x + cos y)

which i can turn into

(1 - sin x) - (1 - sin y) / cos x + cos y + (1 - cos x) (1 - cos y) / (sin x + sin y)

now those fractions cancel out right? cause (1 - sin x) and (1 - sin y) is the same as cos x and cos y as (1 - cos x) and (1 - cos y) are the same as (sin x) and (sin y) right? so if all that cancels out i get 0!

while in the shower it struck me that 1 - sin x = cos x , so i went from there..

i hope that is the right answer, if not could you please help me go in the right direction? thanks for your guys much appreciated help!

 

[hypermorphism]

while in the shower it struck me that 1 - sin x = cos x , so i went from there..

This isn't correct. The equation you may be attempting to reference is sin²(x) + cos²(x) = 1.
Regarding your original equation, try putting the fractions on opposite sides of the equation:
$$\frac{\cos x - \cos y}{\sin x + \sin y} + \frac{\sin x - \sin y}{\cos x + \cos y} = 0$$
is the same as
$$\frac{\cos x - \cos y}{\sin x + \sin y} = -\frac{\sin x - \sin y}{\cos x + \cos y}$$
or
$$\frac{\cos x - \cos y}{\sin x + \sin y} = \frac{\sin y - \sin x}{\cos x + \cos y}$$
Do you see a way to get rid of the fractions from here ?

 

[Data]

Yes, look at hypermorphism's last post for ideas. You seem to have made several errors in simplification.

 

[Integral]

I have bolded your error
(cos^2 x ) - (cos^2 y)/(sin x + sin y) (cos x + cos y) + (sin^2 x)( sin^2 y)/ (sin x + sin y) (cos x + cos y)

go back and double check where this came from, it is not correct. Everything else is ok

 

[InFiNiTeX]

ok thanks for your help Integral! and hypermorphism, sorry i just can't see how to get rid of the fractions... I'm sure its very easy and i will feel really dumb though...

 

[Data]

You have this equation, from hypermorphism's last post:

$$\frac{\cos x - \cos y}{\sin x + \sin y} = \frac{\sin y - \sin x}{\cos x + \cos y}$$

Do you remember cross multiplication?

 

[InFiNiTeX]

OHH! see i told you it would probably be very easy... thanks for your help!