Prove that arctanx + arccotx = pi/2

arctanx + arccotx =
pi/2

We'll associate a function f(x) to the expression
(arctanx + arccotx).

If we have to verify if the function
is a constant function, we'll have to do the first derivative
test. 

When the first derivative of a function is
cancelling, that means that the function is a constant function, because the derivative
of a constant function is 0.

We'll differentiate the
function f(x):

f'(x) = (arctanx +
arccotx)'

f'(x) = 1/(1+x^2) -
1/(1+x^2)

We'll eliminate like
terms:

f'(x)=0,

If
f'(x)=0 => f(x)=constant

Now, we'll determine the
constant for (arctanx + arccotx )' = 0

We'll prove that
the constant is pi/2.

For this reason, we'll put x =
1:

f(1)=arctan1+ arccot1=
pi/2

f(1) = pi/4 + pi/4

f(1) =
2pi/4

f(1) = pi/2
q.e.d.