SOLUTION: cosxtanx=0

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Question 165742: cosxtanx=0
Answer by Edwin McCravy(6938) About Me  (Show Source):
You can put this solution on YOUR website!


cos%28x%29tan%28x%29=0

We have to be careful here!  We might be tempted
to use the zero-factor principle. However that 
will lead us astray because tan%28x%29 is not 
defined at ANY of the values of x for which
cos%28x%29=0!

So do NOT do this:

cross%28matrix%283%2C4%2C+cos%28x%29=0%2C%27%27%2C%27%27%2Ctan%28x%29=0%2C+%0D%0A+++++++++++++++x=pi%2F2%2Bn%2Api%2C%27%27%2C%27%27%2Cx=0%2Bn%2Api%2C%0D%0A++++++++++++++++++++++++%27%27%2C%27%27%2C%27%27%2Cx=n%2Api%29%29%0D%0A+

Instead use the identity matrix%281%2C+3%2Ctan%28alpha%29%2C+%27=%27%2C+sin%28alpha%29%2Fcos%28alpha%29%29 to replace tan%28x%29.

+matrix%281%2C3%2C+cos%28x%29tan%28x%29%2C+%27=%27%2C+0%29+
+matrix%281%2C5%2C+cos%28x%29%2C+%27%2A%27%2C+sin%28x%29%2Fcos%28x%29%2C+%27=%27%2C+0%29

We can cancel cos%28x%29's as long as we are careful 
not to allow any values of x which cause the 
denominator cos%28x%29 to be zero.

+matrix%281%2C5%2C+cross%28cos%28x%29%29%2C+%27%2A%27%2C+sin%28x%29%2Fcross%28cos%28x%29%29%2C+%27=%27%2C+0%29 

That leaves:

matrix%281%2C3%2Csin%28x%29%2C%27=%27%2C0%29

which has solution 

x=n%2Api, where n is any integer.

And this is a valid solution since cos%28n%2Api%29
is always either %27%27%2B1 or -1, and never 0.

Edwin