a-maths!!!

As follows:
By the formula: tan 2A = 2 tan A / ( 1 - tan^2 A ),

圖片參考：http://i238.photobucket.com/albums/ff245/chocolate328154/Maths046.jpg?t=1197901039



2007-12-17 22:50:11 補充：
Something wrong with the answer below. There should NOT be the solutions 90* and 270*. They are only true for L.H.S., i.e. tan 180* and tan 540* are 0 but for R.H.S., they are not valid. tan 90* and tan 270* are undefined and so 8 / undefinded cannot be defined as well.

2007-12-18 13:36:56 補充：
tan ( A + B ) = ( tan A + tan B ) / ( 1 - tan A tan B ), sotan ( A + A ) = ( tan A + tan A ) / ( 1 - tan^2 A )and here comes the formula: tan 2A = 2 tan A / ( 1 - tan^2 A )

2007-12-19 22:56:31 補充：
'If 1/tanz is replaced by cotz, then 90 and 270 degrees are solutions.'It is still wrong becuase 1 / tan z = cot z, if 1 / tan 90* and 1 / tan 270* do not satisfy the equation, then how come cot 90* and cot 270* can satisfy the equation?

Since sin2z=2sinzcosz and cos2z=square(cosz)-square(sinz) and
tanz=sinz/cosz
the equation can be converted to:
sinzsin2z=8coszcos2z
then
2*square(sinz)*cosz=8*cube(cosz)-8cosz*square(sinz)
then by rearranging the terms
5*square(sinz)*cosz=4*cube(cosz)
cosz(5*square(sinz)-4*square(cosz))=0
cosz=0 or tanz=plus or minus(2/squareroot(5))
cosz=0 implies that z= 90 degrees or 270 degrees
tanz=plus or minus(2/squareroot(5)) implies that
z=41.810 or 138.190 or 221.810 or 318.190 degrees, so there are six solutions

2007-12-17 22:26:51 補充：
the two solution 90 degrees and 270 degrees are based on the fact that at these angles,tanz tends to infinity, so 1/tanz is zero (1/tanz=cosz/sinz, cos90degree=cos270degree=0, sin90degree=1,sin270degree= -1)also tan2z are zero (tan180degree and tan 540degree are zero)

2007-12-19 18:37:49 補充：
Sorry, I have made a mistake.cotz= cosz/sinzcotz =0 at 90 and 270 degrees,while 1/tanz is undefined at these angles.so 90 and 270 are not solutions for this equation.However, if 1/tanz is replaced by cotz, then 90 and 270 degrees are solutions.