若函数f(x)=(1+cos2x)/(2sin(π/2-x))+sinx+(a^2)*sin(x+π/4)的最大值为√2+3,试确定常数a的值

如果做错请告诉我.
原式= (1+cos^2x-sin^2x)/2cosx + sinx +a^2 * sin(x+π/4)
=(sin^2x+cos^2x+cos^2x-sin^2x)/2cosx +sinx +a^2 * sin(x+π/4)
=2cos^2x/2cosx +sinx +a^2 * sin(x+π/4)
=cosx+sinx+a^2 * sin(x+π/4)
=√2 sin(x+π/4)+ a^2 * sin(x+π/4) (这一步是合一变型)
=(√2+ a^2 )* sin(x+π/4)

因为sin(x+π/4) Max=1
所以(√2 + a^2 ) =√2+3
所以...... 接下来自己做咯``
打得好累.