Question

（1减去sinx）除于cosx为什么等于tan2分之x

Answer 1

错了。
(1
-sinx)/cosx
=
[
1
-tan
(x/2)
]
/
[
1
+tan
(x/2)
].
应该是：
(1
-cos
x)
/sin
x
=
tan
(x/2),
或：
sin
x
/(1
+cos
x)
=
tan
(x/2).
=
=
=
=
=
=
=
=
=
1.
(1
-cos
x)
/sin
x
=
tan
(x/2).
证明：因为
cos
x
=1
-2
[
sin
(x/2)
]^2,
sin
x
=
2
sin
(x/2)
cos
(x/2),
所以
(1
-cos
x)
/sin
x
=
2
[
sin
(x/2)
]^2
/
[
2
sin
(x/2)
cos
(x/2)
]
=
sin
(x/2)
/
[
cos
(x/2)
]
=
tan
(x/2).
=
=
=
=
=
=
=
=
=
2.
sin
x
/(1
+cos
x)
=
tan
(x/2).
证明：因为
sin
x
=
2
sin
(x/2)
cos
(x/2),
cos
x
=
2
[
cos
(x/2)
]^2
-1,
所以
sin
x
/(1
+cos
x)
=
...
=tan
(x/2).
=
=
=
=
=
=
=
=
=
同理,
(1
+cos
x)
/sin
x
=sin
x
/(1
-cos
x)
=
cot
(x/2).
cos
x
=
[
cos
(x/2)
]^2
-[
sin
(x/2)
]^2
=
1
-2
[
sin
(x/2)
]^2
=
2
[
cos
(x/2)
]^2
-1,
公式的选取很关键.