Question

If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =

(a) −1
(b) $\frac{1}{2}$
(c) $-\frac{1}{2}$
(d) None of these

Answer 1

(c)$\frac{-1}{2}$

$$\begin{gathered} \tan 135°=\tan (90°+45°) \\ =-\tan 45° \\ =-1 \\ \mathrm{Or},\tan (69°+66°)=\frac{\tan 69°+\tan 66°}{1-\tan 69°\tan 66°} \\ \Rightarrow -1=\frac{\tan 69°+\tan 66°}{1-\tan 69°\tan 66°} \\ \Rightarrow \tan 69°+\tan 66°-\tan 69°\tan 66°=-1 \\ \mathrm{Therefore}, \\ 2k\mathit{=}-1 \\ \mathit{\Rightarrow }k\mathit{=}\frac{-1}{2} \end{gathered}$$