Question

数学题：很简单的转化吧！

已知：x+y=√2sin(a+π/4), x-y= √2sin(a-π/4).
求证：x^2+y^2=1

Answer 1

（x+y)^2+（x-y）^2=2sin(a+π/4)^2+2sin(a-π/4)^2=2sin(a+π/4)^2+2cos(a-π/4+π/2)^2=2

=>x^2+y^2=1

Answer 2

证明
2x=√2（sin(a+π/4)+sin(a-π/4))=√2(sin(a+π/4)-cos(a+π/4))
2y=√2(sin(a+π/4)+cos(a+π/4))
x^2+y^2=1/2(sin(a+π/4)-cos(a+π/4))^2+1/2(sin(a+π/4)+cos(a+π/4))^2
=1/2(1-sin2a)+1/2(1+sin2a)=1

Answer 3

对已知条件进行平方：X^2+2XY+Y^2=2SIN(A+π/4) (1)
X^2-2XY+Y^2=2SIN(A-π/4) (2)
(1)+(2):2X^2-2Y^2=2{sin(a+π/4)+sin(a+π/4)}
化简：x^2+y^2=1