Question

$\mathrm{If}\tan 70°=a\tan 50˚+b\tan 20˚\mathrm{then}\mathrm{ab}\mathrm{is}\mathrm{equal}\mathrm{to}$.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is B 2

Here, by using the expression $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$
we get, $\tan 70°=\tan (50°+20°)\Rightarrow \tan 70°=\frac{\tan 50°+\tan 20°}{1-\tan 50°\tan 20°}\Rightarrow \tan 70°\left(1-\tan 50°\tan 20°\right)=\tan 50°+\tan 20°\Rightarrow \tan 70°-\tan 70°\tan 50°\tan 20°=\tan 50°+\tan 20°\Rightarrow \tan 70°=\tan 70°\tan 50°\tan 20°+\tan 50°+\tan 20°$
Also we know $\tan 70°=\tan (90°-20°)\Rightarrow \tan 70°=cot20°$
Thus we get,
$\tan 70°=cot20°\tan 50°\tan 20°+\tan {50}^o+\tan 20°\Rightarrow \tan 70°=2\tan 50°+\tan 20°$
Now, comparing we get,
$a=2,b=1$
⇒ab=2.