M2 Prove Trigo

L.H.S.

=

1 + csc(270 + x) - tan(x - 180)
--------------------------------------------
1 - csc(x - 90) + tan(x + 180)


=

1 - sec x - tan x
------------------------
1 + sec x + tanx
=

1 - 1/cosx - sinx / cosx
-----------------------------------
1 + 1/cosx + sinx / cosx=

cosx - sinx - 1
-----------------------
cosx + sinx + 1
=

(cosx - sinx - 1)(cosx + sinx + 1)
------------------------------------------------
(cosx + sinx + 1)²
=

cos²x - (sin²x + 2sinx + 1)
------------------------------------------------------------------------
sin² + cos²x + 2sinx cosx + 2(cosx + sinx) + 1
=

- 2sin²x - 2sinx
---------------------------------------------
2(1 + sinx cosx + cosx + sinx)
=

- 2sinx (sinx + 1)
------------------------------
2(cosx + 1)(sinx + 1)
=

- sinx
--------------
cosx + 1
=

(- sinx)(cosx - 1)
-----------------------------
(cosx + 1)(cosx - 1)
=

(- sinx)(cosx - 1)
-------------------------
- sin²x
=

cosx - 1
-------------
sinx
=

(cosx - 1)/cosx
------------------------
sinx / cosx
=

1 - sec x
-------------
tanx
= R.H.S.


2011-01-10 13:59:41 補充：
The steps after

cosx - sinx - 1
----------------------- can be more simple :
cosx + sinx + 1

=

(cosx - sinx - 1)²
-------------------------------------------------
(cosx + sinx + 1)(cosx - sinx - 1)

=