Question

The product $(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })\dots (1+\tan 45^{\circ })$ equals

• A. $2^{21}$
• B. $2^{22}$
• C. $2^{23}$
• D. $2^{25}$

Answer 1

Answer: (C)

We have,
$(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })\dots (1+\tan 45^{\circ })$
We know that,
$(1+\tan \theta )(1+\tan (45^{\circ }-\theta ))=2$
$\therefore (1+\tan 1^{\circ })(1+\tan 44^{\circ })(1+\tan 2^{\circ })$
$(1+\tan 43^{\circ })\dots (1+\tan 22^{\circ })(1+\tan 23^{\circ })$
$(1+\tan 45^{\circ })$
$\Rightarrow 2^{22}.2=2^{23}$