Question

The value of $(1+\tan 1^{\circ })(1+\tan 2^{\circ })\dots (1+\tan {45}^{\circ })$ is

• A. $2^{21}$
• B. $2^{22}$
• C. $2^{23}$
• D. $2^{24}$

Answer 1

The correct option is C $2^{23}$

We know that,
$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$
Assuming $A+B={45}^{\circ }$
$\tan A+\tan B=1-\tan A\tan B\Rightarrow \tan A+1+\tan B(\tan A+1)=2$
$\Rightarrow (1+\tan A)(1+\tan B)=2$
Now,
$(1+\tan 1^{\circ })(1+\tan 2^{\circ })\dots (1+\tan {45}^{\circ })=(1+\tan 1^{\circ })(1+\tan {44}^{\circ })(1+\tan 2^{\circ })(1+\tan {43}^{\circ })\dots (1+\tan {45}^{\circ })=2^{23}$