Question

The smallest positive value of X ( in degree ) for which $tan(x+{100}^{\circ })=tan(x+{50}^{\circ })tanXtan(x-{50}^{\circ })$ is

• A. 30
• B. 45
• C. 60
• D. 55

Answer 1

The correct option is A 30

$tan(x+100)=tan(x+50)\times tan\left(x\right)\times tan(x-50)$

$\frac{tan(x+100)}{tan(x-50)}=tan(x+50)tan\left(x\right)$

$\frac{sin(x+100)cos(x-50)}{cos(x+100)sin(x-50)}=\frac{sin(x+50)sinx}{cos(x+50)cosx}$

$=\frac{sin(2x+50)}{sin150}=\frac{cos50}{-cos(2x+50)}$

$sin(2x+50)cos(2x+50)=-sin150cos50$

$2sin(2x+50)cos(2x+50)=-cos50$

$sin(4x+100)=sin(270-50)$

$sin(4x+100)=sin220$

$4x+100=220$

$x=30$