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Question

Value of $$\tan x + \tan z$$ is equal to


A
3
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B
0
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C
4
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D
2
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Solution

The correct option is C $$4$$
$$\sum x = \dfrac{3\pi}{4}, \sum \tan x = 5, \tan x \tan y \tan z = 1$$
Using the expansion of $$\tan(x+y+z),$$ 
$$\tan(x+y+z) = \dfrac{\sum \tan x - \tan x \tan y \tan z}{1 - \tan x \tan y - \tan x \tan z - \tan y \tan z}$$
Substituting the given values we get,
$$\tan x \tan y + \tan x \tan z + \tan y \tan z = 5$$
Let $$\tan x  = a, \tan y = b, \tan z = c$$
It can be observed that $$a,b,c$$ are roots of the equation,
$$m^3 - 5m^3 + 5m^2 - 1 = 0$$
Roots of which are,
$$1,2\pm\sqrt3$$
As $$x < y < z$$ and $$\tan$$ is increasing function
$$\tan x = 2 - \sqrt3, \tan y = 1 , \tan z = 2+\sqrt3$$
Hence, $$\tan x + \tan z = 4$$

Mathematics

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