题目

x+y=3-cos4θ,x-y=4sin2θ,求证x^1/2+y^1/2=2
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提问时间：2021-09-09

答案

因为
cos(4t) = 1 - 2 sin(2t)sin(2t)
所以
x+y = 2 + 2 sin(2t)sin(2t)
x-y = 4 sin(2t)
两式分别相加和相减得到
x = 1 + 2 sin(2t) + sin(2t)sin(2t) = ( 1 + sin(2t) ) ^ 2
y = 1 - 2 sin(2t) + sin(2t)sin(2t) = ( 1 - sin(2t) ) ^ 2
开方以后得到
x^(1/2) = 1 + sin(2t)
y^(1/2) = 1 - sin(2t)
所以
x^(1/2) + y^(1/2) = 2