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