高一数学急急急已知sin(α+π/2)=-根号5/5,α∈(0,π),求cos²(π/4+α/2)-cos²(π/4-α/2)/sin(π-α)+cos(3π+α)的值

问题描述:

高一数学急急急已知sin(α+π/2)=-根号5/5,α∈(0,π),求cos²(π/4+α/2)-cos²(π/4-α/2)/sin(π-α)+cos(3π+α)的值

sin(α+π/2)
=cosα
=-√5/5
∴α∈(π/2,π)
sinα=2√5/5
[cos²(π/4+α/2)-cos²(π/4-α/2)]/[sin(π-α)+cos(3π+α)]
=[cos(π/4+α/2)+cos(π/4-α/2)][cos(π/4+α/2)-cos(π/4-α/2)]/(sinα-cosα)
=√2cos(α/2)*(-√2sin(α/2))/(sinα-cosα)
=-2sin(α/2)cos(α/2)/(sinα-cosα)
=-sinα/(sinα-cosα)
=-2/3