Where is tan(xy) continuous?

Question

Where is tan(xy) continuous? I know that it is odd multiples of pi/2, but what is the reasoning behind that?

 

Any help would be much appreciated!

Answer 1

For analysis-type problems like this, it's always always always easiest to think of tan(x) as sin(x)/cos(x).

 

Now we can say that tan(x) is continuous everywhere that it does not become undefined. That is, tan(x) is continuous as long as its denominator is not equal to zero. To find the discontinuous points, just set the denominator to zero and solve:

 

cos(x)=0

 

x= n*pi/2 for n=odd.

 

Similarly, you can very easily find the zeros of tan(x) by setting its numerator to zero and solving. For any kind of analysis of trig functions (tan, cot, sec, csc), your very first instinct should be to express everything in terms of sines and cosines.