Infinity times Zero equals Two

[AntonVrba]

Something to befuddle your minds with
(however, do not believe everything I say)

One can very easily set up the following trigenomical identity:

(1+Tan^2 A)(1+Cos 2A) = 2 … (1)
(one plus tan squared A)...

and its conjugate

(1+Cot^2 B)(1-Cos 2B)=2 …(2)

when A->pi/2 (90 degrees) or B->0
then (1+Tan^2 A) approaches infinity
and (1+Cos 2A) approaches zero ( Cos pi = -1)
but (1) is true hence infinity times zero equals two

Try it yourself, identities (1) and (2) are true

Now you have the fools-proof that:
Infinity times Zero equals Two.

[retina]

Originally Posted by AntonVrba

(1+Tan^2 A)(1+Cos 2A) = 2 … (1)

Of course you left out one important detail: tan(pi/2) is undefined (just like the text beneath my handle)

[science_man_88]

Originally Posted by retina

Of course you left out one important detail: tan(pi/2) is undefined (just like the text beneath my handle)

PARI gives me

1/2*PI + 1/24*PI^3 + 1/240*PI^5 + 17/40320*PI^7 + 31/725760*PI^9 + 691/159667200*PI^11 + 5461/12454041600*PI^13 + 929569/20922789888000*PI^15 + O(PI^17)

[10metreh]

Originally Posted by science_man_88

PARI gives me

1/2*PI + 1/24*PI^3 + 1/240*PI^5 + 17/40320*PI^7 + 31/725760*PI^9 + 691/159667200*PI^11 + 5461/12454041600*PI^13 + 929569/20922789888000*PI^15 + O(PI^17)

It should be Pi not PI. But even then I get -3.961408125 E28, and this number tends to -infinity as realprecision in Pari tends to +infinity. The same thing happens with tan(Pi) (except the number is very small, tending to 0 as realprecision tends to infinity). What's going on here?

[science_man_88]

Originally Posted by 10metreh

It should be Pi not PI. But even then I get -3.961408125 E28, and this number tends to -infinity as realprecision in Pari tends to +infinity. The same thing happens with tan(Pi) (except the number is very small, tending to 0 as realprecision tends to infinity). What's going on here?

then this happens

(11:13) gp > tan(Pi/2)
  *** tan: can't compute tan(Pi/2 + kPi).