Real numbers $x$ and $y$ satisfy the system of equalities
$$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$Prove that $\cos 2x = \cos 2y$.
parmenides51 wrote:
Real numbers $x$ and $y$ satisfy the system of equalities
$\sin x + \cos y = 1$
$\cos x + \sin y = -1$.
Prove that $\cos 2x = \cos 2y$.
( sin x + cos y)²=1
sin²x+2sin x cos y+cos²y=1
(cos x+ sin y)²=-1²
cos²x+2cos x sin y+ sin²y=1
Add,
2(sin x cosy+cos x sin y)=0
2sin(x+y)=0
2sin(x+y) sin(x-y)=0
cos 2x-cos 2y=0
cos 2x=cos 2y( proved)
@Krishijivi
