Question on proving trigonometric identities i found on reddit

[Horny Panda]

If tan(theta) + sin(theta) = x and
tan(theta) - sin(theta) = y,

prove that x⁴ + y⁴ = 2xy(8 +xy)


i tried doing it myself but ended up with a heck ton of work and i still couldn't prove it.
Counting on your help.

 

[MarkFL]

If we add the two given equations, we obtain:

[MATH]\tan(\theta)=\frac{x+y}{2}[/MATH]
And, if we subtract the latter from the former, we obtain:

[MATH]\sin(\theta)=\frac{x-y}{2}[/MATH]
If we multiply the two given equations, we ultimately obtain:

[MATH]\sin^2(\theta)\tan^2(\theta)=xy[/MATH]
Using the two results above, this implies:

[MATH]((x-y)(x+y))^2=16xy[/MATH]
Can you proceed?

 

[Steven G]

Mark, you lost an important parenthesis. To op, I suspect the last equation should read \(\displaystyle ((x-y)(x+y))^2 = 16xy\)

 

[MarkFL]

Mark, you lost an important parenthesis. To op, I suspect the last equation should read \(\displaystyle ((x-y)(x+y))^2 = 16xy\)

I guess you saw my post before I edited it...'twas a tiny window but you found it open.