Question

$\frac{\tan {80}^{\circ }-\tan {10}^{\circ }}{\tan {70}^{\circ }}$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

$\frac{\tan 80°-\tan 10°}{\tan 70°}$

$\Rightarrow \frac{\frac{\sin 80°}{\cos 80°}-\frac{\sin 10°}{\cos 10°}}{\frac{\sin 70°}{\cos 70°}}=\frac{\frac{\left(\sin 80°\cos 10°-\sin 10°\cos 80°\right)}{\cos 80°\cos 10°}}{\frac{\sin 70°}{\cos 70°}}$

$=\frac{\sin (80°-10°).\cos 70°}{\cos 80°\cos 10°\sin 70°}$

$=\frac{\sin 70°\cos 70°}{\cos 80°\cos 10°\sin 70°}$

$=\frac{\cos 70°}{\cos 80°\cos 10°}$

$=\frac{\cos (80°-10°)}{\cos 80°\cos 10°}$

$=\frac{\cos 80°\cos 10°+\sin 80°\sin 10°}{\cos 80°\cos 10°}$

$=1+\tan 80°\tan 10°$

$=1+\tan (90°-10°)\tan 10°$

$=1+\cot 10°\tan 10°$

$=1+1$

$=2.$

Hence, the answer is $2.$