As #tan^2x-6tanx+5=0# is a quadratic equation in #tanx#, let us first factorize it to find a solution #for #tanx#.
#tan^2x-6tanx+5=0# is equivalent to
#tan^2x-tanx-5tanx+5=0#
or #tanx(tanx-1)-5(tanx-1)-0#
or #(tanx-5)(tanx-1)=0#
i.e. #tanx=5# or #tanx=1#
As #tan^(-1)p# indicates an angle, say #theta#, whose tangent is #p# i.e. #tantheta=p#
In the given question, we are not to find #tanx# but #x# and hence we use definition of inverse function for this and solution of given equation is
#x=tan^(-1)1# or #x=tan^(-1)5#
It is apparent that #tan^(-1)1=p/4# but it is not so easy for #tan^(-1)5#.
To find exact value of #x#, we need to either look at inverse function tables or use scientific calculator and using this, we get the value of #x# (in radians as it is an angle) is
#x=pi/4=0.7854# or #x=1.3734#
It is apparent that #pi+pi/4=(5pi)/4=3.927# and #pi+1.3734=4.515#
