Are there any other values of a such that the linearization of sin(x) at a = the linearization of tan(x) at a? Describe them through proof if they exist. If they don't exist, proove that instead.

Question

Are there any other values of a such that the linearization of sin(x) at a = the linearization of tan(x) at a?  Describe them through proof if they exist.  If they don't exist, proove that instead.

phi · Answer

FIrst, write down the linearization  for both functions, using

$$ f_{a1}(x) \approx f_1(a)+ f_1'(a)(x-a) \ f_{a2}(x) \approx f_2(a)+ f_2'(a)(x-a)$$
then equate terms
$$ f_1(a) = f_2(a) \ f_1'(a) = f_2'(a) $$
and solve for a

anonymous · Answer

how do you solve for a?