灵鹊做题网求证：2sinx•cosx(sinx+cosx−1)(sinx−cosx+1)＝1+cosxsinx
来源：学生作业帮 编辑：灵鹊做题网作业帮 分类：数学作业 时间：2024/04/26 03:33:20
求证：
＝
| 2sinx•cosx |
| (sinx+cosx−1)(sinx−cosx+1) |
| 1+cosx |
| sinx |
左边=
2sinx•cosx
(sinx+cosx−1)(sinx−cosx+1)
=
2sinx•cosx
sin2x−(cosx−1)2
=
2sinx•cosx
sin2x−cos2x+2cosx−1
=
2sinx•cosx
−2cos2x+2cosx
=
sinx
1−cosx
=
sinx(1+cosx)
(1−cosx)(1+cosx)
=
sinx(1+cosx)
1−cos2x
=
1+cosx
sinx=右边．
∴
2sinx•cosx
(sinx+cosx−1)(sinx−cosx+1)＝
1+cosx
sinx．
2sinx•cosx
(sinx+cosx−1)(sinx−cosx+1)
=
2sinx•cosx
sin2x−(cosx−1)2
=
2sinx•cosx
sin2x−cos2x+2cosx−1
=
2sinx•cosx
−2cos2x+2cosx
=
sinx
1−cosx
=
sinx(1+cosx)
(1−cosx)(1+cosx)
=
sinx(1+cosx)
1−cos2x
=
1+cosx
sinx=右边．
∴
2sinx•cosx
(sinx+cosx−1)(sinx−cosx+1)＝
1+cosx
sinx．
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