Question

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

• A. $3$
• B. $0$
• C. $4$
• D. $2$

Answer 1

The correct option is C $4$

$\sum x=\frac{3\pi }{4},\sum \tan x=5,\tan x\tan y\tan z=1$

Using the expansion of $\tan (x+y+z),$

$\tan (x+y+z)=\frac{\sum \tan x-\tan x\tan y\tan z}{1-\tan x\tan y-\tan x\tan z-\tan y\tan z}$

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,

$1,2\pm \sqrt{3}$

As $x<y<z$ and $\tan$ is increasing function

$\tan x=2-\sqrt{3},\tan y=1,\tan z=2+\sqrt{3}$

Hence, $\tan x+\tan z=4$