Solve : `2+tan x. cot.(x)/(2)+cot x. tan.(x)/(2)=0`.

Answer

`x = 2n pi pm(2pi)/(3), n in Z`

Explanation

`2+tan x.cot. (x)/(2)+cotx.tan.(x)/(2)=0` …(1)
Let `an x.cot.(x)/(2)=y`
Then equation (1) becomes
`2+(y+(1)/(y))=0`
`rArr y+(1)/(y)=-2`
On solving, we get
y = -1
or `tan x. cot.(x)/(2)=-1`
or `(sin x)/(cos x)(cos.(x)/(2))/(sin.(x)/(2))=-1`
or `(2sin.(x)/(2)cos.(x)/(2))/(cos x).(cos.(x)/(2))/(sin.(x)/(2))-1`
or `(2"cos"^(2)(x)/(2))/(cos x)=-1`
or `(1+cos x)/(cos x)=-1`
or `cos x= -(1)/(2)`
or `x=2n pi pm (2pi)/(3), n in Z`