若arcsin(sinA + sin B) + arcsin(sinA - sinB) = π /2,求 (sin²A + sin²B) 的值.

问题描述:

若arcsin(sinA + sin B) + arcsin(sinA - sinB) = π /2,求 (sin²A + sin²B) 的值.

令P=arcsin(sinA + sinB),Q=arcsin(sinA - sinB).则sinP = sinA + sinB①, sinQ = sinA - sinB②
由题设,sinP=cosQ,则sin^2P=cos^2Q=1-sin^2Q,
sin^2P+sin^2Q=1.
①~②带入,可得:sin²A + sin²B = 1/2