已知α和β是锐角,且sin^2(α)+sin^2(β)=sin(α+β),求证α+β=π/2

Question

已知α和β是锐角,且sin^2(α)+sin^2(β)=sin(α+β),求证α+β=π/2

Answer 1

已知:sin^2(α)+sin^2(β)=sin(α+β) (1)
为方便,用a,b,记α,β.
a,b为锐角时,pi/2-a,pi/2-b也是锐角,故
由(1),也就有:sin^2(pi/2-α)+sin^2(pi/2 -β)=sin[pi-(α+β)]
即:cos^2(α)+cos^2(β)=sin(α+β) (2)
(1)+(2),得:
2=2sin(α+β),
sin(α+β)=1,
即得:sin(α+β)=pi/2