【答案】
分析:(Ⅰ)(法一)由比例性质(1-cosx)•(1+cosx)=1-cos
2x=sin
2x可证;
(法二)利用sin
2x+cos
2x=1,移项整理即可;
(法三)作差整理,最后证得差为0即可.
(Ⅱ)利用诱导公式与三角函数间的关系式即可证得结论.
解答:(Ⅰ)证明:(法一)利用比例性质
∵(1-cosx)•(1+cosx)=1-cos
2x=sin
2x
∴
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/0.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/1.png)
…(5分)
(法二)
∵sin
2x+cos
2x=1,
∴1-cos
2x=sinx•sinx,即(1-cosx)•(1+cosx)=sinx•sinx
又∵(1-cosx)≠0,sinx≠0
∴
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/2.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/3.png)
…(5分)
(法三)
∵
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/4.png)
-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/5.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/6.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/7.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/8.png)
=0
∴
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/9.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/10.png)
…(5分)
(Ⅱ)原式=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/11.png)
+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/12.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/13.png)
+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/14.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/15.png)
-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/16.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/17.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103103041305664826/SYS201311031030413056648020_DA/18.png)
=1.…(12分)
点评:本题考查三角函数恒等式的证明,着重考查诱导公式与同角三角函数间的基本关系,考查三角函数的化简求值,属于中档题.