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