Question

化简 (1+sinα-cosα)/(1+sinα+cosα)+(1+sinα+cosα)/(1+sinα-cosα)

求详细过程 ！

Answer 1

∵(1+sinα-cosα)/(1+sinα+cosα)
=(1-cosα+sinα)/(1+cosα+sinα)
=(2sin²(α/2)+sinα)/(2cos²α/2+sinα)
=(2sin²α/2+2sinα/2cosα/2)/(2cos²α/2+2sinα/2cosα/2)
=(sin²α/2+sinα/2cosα/2)/(cos²α/2+sinα/2cosα/2)
=[sinα/2(sinα/2+cosα/2)]/[cosα/2(cosα/2+sinα/2)]
= sinα/2/ cosα/2

∴(1+sinα-cosα)/(1+sinα+cosα)+(1+sinα+cosα)/(1+sinα-cosα)
= sinα/2/ cosα/2+ cosα/2/ sinα/2
= [sin²α/2 cos²α/2]/ [cosα/2 sinα/2]
=1/[cosα/2 sinα/2]
=2/[2cosα/2 sinα/2]
=2/ sinα.