Arctg(x) + Arcctg(1/3) = Pi/2. Find the x value.

You can answer this question by recalling (or looking up) an obscure trigonometric identity (see the link):

tan(x) + atan(1/x) = pi/2 if x > 0

.                             = -pi/2 if x < 0

From the second term (1/x = 1/3), we know that x = 3

To prove this identity, we can make use of well known pi/2 phase shift identities of trig functions, in this case tan(y) = cot(pi/2 - y)

Let y = atan(x);   then x = tan(y) = cot(pi/2 - y)

x = cot(pi/2 - y)

acot(x) = acot(cot(pi/2 - y)) = pi/2 - y = pi/2 - atan(x)

--> pi/2 = atan(x) + acot(x)

This form is more well known than the first form given above. To get to that form, you have to recall that,

acot(x) = atan(1/x) , x > 0

acot(x) = -pi + atan(1/x) , x < 0