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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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