已知锐角θ满足sin(πcosθ)=cos(πsinθ) 求sin2θ的值?

要使sin(πcosθ)=cos(πsinθ)的条件满足,则有两种情况①和②
① πcosθ + πsinθ = π/2
即 cosθ + sinθ = 1/2
即 1/4 = (cosθ + sinθ)^2 = 1 + 2sinθcosθ = 1 + sin2θ
得：sin2θ = -3/4
因为θ是锐角,故sin2θ应大于0,因此这种情况不满足条件
② πcosθ - πsinθ = π/2
即 cosθ - sinθ = 1/2
即 1/4 = (cosθ - sinθ)^2 = 1 - 2sinθcosθ = 1 - sin2θ
得：sin2θ = 3/4 满足sin2θ＞0的条件
综上所述：sin2θ = 3/4