Recall that #y = tanx# can be written as #y = sinx/cosx#. Then there will be asymptotes whenever #cosx = 0#, since we cannot have the denominator equal #0# without making the function undefined in the real number system.

Similarly, #f(x) = tan2x# can be rewritten as #f(x) = (sin2x)/(cos2x)#. We need to find the values of #x# that make #cos2x= 0#.

#cos2x= 0#

#2x = arccos(0)#

#2x = pi/2, (3pi)/2#

#x = pi/4 and (3pi)/4#

Don't forget to add the periodicity. The function #y = cos2x# has a period of #pi#, so there will be asymptotes in #tan(2x)# whenever #x= pi/4 + pin# or #x = (3pi)/4 + pin#, where #n# is an integer.

Hopefully this helps!