If cot-1(α) + cot-1(β) = cot-1(x), then x

1) α + β

2) α – β

3) (1 + αβ)/(α + β)

4) (αβ – 1)/(α + β)

Answer: (4) (αβ – 1)/(α + β)

Solution:

Given,

cot-1(α) + cot-1(β) = cot-1(x)

tan-1(1/α) + tan-1(1/β) = tan-1(1/x)

Using the formula tan-1(a) + tan-1(b) = tan-1[(a + b)/(1 – ab)],

tan-1[{(1/α) + (1/β)}/ {1 – (1/α)(1/β)}] = tan-1(1/x)

tan-1[(β + α)/(αβ – 1)] = tan-1(1/x)

Thus, (α + β)/(αβ – 1) = 1/x

x = (αβ – 1)/(α + β)

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