Question 966873
by definition, tan(a) = b if and only if arctan(b) = a


therefore:


if tan(a) = b, then arctan(b) = a


tan(arctan(b)) is therefore equal to tan(a) which is equal to b.


this gets you tan(arctan(b)) = b.


if you let b = 41/15, then you get:


tan(arctan(b)) = b becomes tan(arctan(41/15)) = 41/15.


that takes care of your first statement.


now to move on to your second statement.


by definition, arctan(c) = d if and only if tan(d) = c.


therefore, if arctan(c) = d, then tan(d) = c.


arctan(tan(d) is therefore equal to arctan(c) which is equal to d.


this gets you arctan(tan(d)) = d.


if you let d = 70pi/56, then you get:


arctan(tan(70pi/56)) = 70pi/56.


you can confirm these properties are correct by using your calculator.
set your calculator to degree mode.


tan(45) = 1
arctan(1) = 45


this leads to:


tan(arctan(1)) = 1
arctan(tan(45) = 45