Question

If 2 tan x 1- tan2 x 1 V3 then x can equal: + MT B. X= A. x=112 x- 7 C. X = ST D. X=

Answer 1

Solve for x

`(2\tan (x))/(1-tan^2(x))=\frac{1}{\sqrt[]{3}}`

Step 1: Let tan(x) represent y

`\begin{gathered} (2y)/(1-y^2)=\frac{1}{\sqrt[]{3}} \\ \text{cross multiply} \\ 1-y^2=2\sqrt[]{3}y \\ y^2+2\sqrt[]{3}y-1=0 \end{gathered}`

Step 2: Using quadratic equation solve for y

`\begin{gathered} y^2+2\sqrt[]{3}y-1=0 \\ y=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ a=1,b=2\sqrt[]{3},c=-1 \\ y=\frac{-2\sqrt[]{3}\pm\sqrt[]{(2\sqrt[]{3})^2-4(1)(-1)}}{2(1)} \\ y=\frac{-2\sqrt[]{3}\pm\sqrt[]{12+4}}{2} \\ y=\frac{-2\sqrt[]{3}\pm\sqrt[]{16}}{2} \\ y=\frac{2(-\sqrt[]{3}\pm2)}{2} \\ y=-\sqrt[]{3}\pm2 \end{gathered}`

Since the y represents tan x

`\begin{gathered} \tan \mleft(x\mright)=-√(3)-2,\: \tan \mleft(x\mright)=2-√(3) \\ x=\arctan \mleft(-√(3)-2\mright)+\pi n,\: x=\arctan \mleft(2-√(3)\mright)+\pi n \\ x\text{ in degrees} \\ \: x=-75^(\circ\: )+180^(\circ\: )n,\: x=15^(\circ\: )+180^(\circ\: )n \\ \: x=15^(\circ\: )+180^(\circ\: )n \end{gathered}`

Therefore the correct answer is

`x=(\pi)/(12)+n\pi`

Hence the correct answer is Option B