若8cos(pi/4+a)cos(pi/4-a)=1,则sina^4+cosa^4=_________.

题目

若8cos(pi/4+a)cos(pi/4-a)=1,则sina^4+cosa^4=_________.

答案

因为(PI/4 + a) + (PI/4 - a) = PI/2
也即PI/4 + a = PI/2 - (PI/4 - a)
所以cos(PI/4 + a) = sin(PI/4 - a)
所以
8cos(pi/4+a)cos(pi/4-a)
= 8 * sin(PI/4 - a) * cos(PI/4 - a)
= 4sin(PI/2 - 2a)
= 4cos(2a) = 1
故cos2a = cosa^2 - sina^2 = 1/4
因为cosa^2 + sina^2 = 1
所以cosa^2 = 5/8,sina^2 = 3/8
所以sina^4 + cosa^4 = (cosa^2 + sina^2)^2 - 2(sina^2)(cosa^2)
= 1 - 2 * 5/8 * 3/8 = 1 - 30/64 = 34/64 = 17/32
希望有用.