Question

If $(1 + \tan \, 1^{\circ})(1 + \tan \, 2^{\circ}) ...... (1 + \tan \, 45^{\circ}) = 2^n$, then n is

• A. 22
• B. 24
• C. 23
• D. 12

Answer 1

Answer: (C)

We have,
$\left(1+\tan 1^{\circ}\right)\left(1+\tan 2^{\circ}\right) \ldots .\left(1+\tan 45^{\circ}\right)=2^{ n }$
Now, $(1+\tan \theta)\left(1+\tan \left(45^{\circ}-\theta\right)\right)$
$=(1+\tan \theta)\left(1+\frac{1-\tan \theta}{1+\tan \theta}\right)$
$=(1+\tan \theta) \frac{(1+\tan \theta+1-\tan \theta)}{1+\tan \theta}=2$
$\therefore \left(1+\tan ^{\circ}\right)\left(1+\tan 44^{\circ}\right)\left(1+\tan 2^{\circ}\right)\left(1+\tan 43^{\circ}\right)$
$\ldots\left(1+\tan 22^{\circ}\right)\left(1+\tan 23^{\circ}\right)\left(1+\tan 45^{\circ}\right)$
$=2\times2\times2\ldots\ldots\ldots 22$ times $\times \left(1+1\right)$
$=2^{22}\times2=2^{23}$
$\therefore n=23$