Evaluate the value of this expression: arctan (1/3)+arctan(1/5)+arctan(1/7)+arctan(1/8)

First, we'll note this expression by
E.

E = arctan
(1/3)+arctan(1/5)+arctan(1/7)+arctan(1/8)

We'll put arctan
(1/3) = a and arctan (1/5) = b.

We'll subtract
arctan(1/7)+arctan(1/8) both sides:

E -
(arctan(1/7)+arctan(1/8)) = a + b

We'll also note
arctan(1/7) = c and arctan(1/8) = d

E - (c+d) = a +
b

We'll apply tangent function both
sides:

tan [E - (c+d)] =
tan(a+b)

We'll recall the
identity:

tan(a+b) = (tan a + tan b)/(1 - tan a*tan
b)

But tan(arctan x) =
x.

Accordingly to this identity, we'll
have:

tan[arctan (1/3)+arctan(1/5)] = (1/3 + 1/5)/(1 -
1/15)

tan[arctan (1/3)+arctan(1/5)] =
8/14

tan[arctan (1/3)+arctan(1/5)] =
4/7

tan  [E - (c+d)] = [tan E - tan
(c+d)]/[1-tanE*tan(c+d)]

Tan(c+d) = (1/7 + 1/8)/(1 +
1/56)

Tan(c+d) =
15/55

Tan(c+d) = 3/11

tan  [E
- (c+d)] = [tan E - 3/11]/[1-tanE*3/11]

tan  [E - (c+d)] =
(11tan E - 3)/(11 + 3tan E)

The expresison will
become:

(11tan E - 3)/(11 + 3tan E) =
4/7

7(11tan E - 3) = 4(11 + 3tan
E)

We'll remove the
brackets:

77tanE - 21 = 44 +
12tanE

We'll move the unknown terms to the
left:

77tanE - 12tanE = 21 +
44

65tanE = 65

We'll divide by
65:

tan E = 1

E =arctan
1

E = pi/4

The
value of the expression E = arctan (1/3)+arctan(1/5)+arctan(1/7)+arctan(1/8) is E =
pi/4.