Question

sinA+sin(90°-A)+1/sinA-sin(90°-A)+1=cot(A/2) 请问这怎么证明,

Answer 1

证明：∵sinA+sin(90°-A)+1=sinA+cosA+1

=2sin(A/2)cos(A/2)+2[cos(A/2)]^2

=2cos(A/2)[sin(A/2)+cos(A/2)]

sinA-sin(90°-A)+1=sinA-cosA+1

=2sin(A/2)cos(A/2)+2[sin(A/2)]^2

=2sin(A/2)[cos(A/2)+sin(A/2)]

∴[sinA+sin(90°-A)+1]/[sinA-sin(90°-A)+1]=[2cos(A/2)]/[2sin(A/2)]=cot(A/2)

∴[sinA+sin(90°-A)+1]/[sinA-sin(90°-A)+1]=cot(A/2)