Question

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

A
3
B
0
C
4
D
2

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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