Question

If $tan{69}^{\circ }+tan{66}^{\circ }-tan{69}^{\circ }tan{66}^{\circ }=2k,$ then k=

• A. -1
  
• B. $\frac{1}{2}$
  
• C. $-\frac{1}{2}$
  
• D. none of these

Answer 1

The correct option is C $-\frac{1}{2}$


$tan{135}^{\circ }=tan({90}^{\circ }+{45}^{\circ }$
= $-tan{45}^{\circ }$
=-1
$tan({69}^{\circ }+{66}^{\circ })=\frac{tan{69}^{\circ }+tan{66}^{\circ }}{1-tan{69}^{\circ }tan{66}^{\circ }}$
$\Rightarrow -1=\frac{tan{69}^{\circ }+tan{66}^{\circ }}{1-tan{69}^{\circ }tan{66}^{\circ }}$
$\Rightarrow tan{69}^{\circ }+tan{66}^{\circ }-tan{69}^{\circ }tan{66}^{\circ }=-1$
Therefore,
2k=-1
$\Rightarrow k=\frac{-1}{2}$