Question

If $(1+\tan 1^{\circ })\cdot (1+\tan 2^{\circ })\cdot (1+\tan 3^{\circ })\dots \cdot (1+\tan {45}^{\circ })=2^n$ , then ${}^{\prime }{n}^{\prime }$ is equal to ?

• A. 16
• B. 23
• C. 30
• D. None of these

Answer 1

The correct option is B 23

Given

$(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })\cdots (1+\tan {45}^{\circ })$

Let us Consider two terms

$(1+\tan x)(1+\tan (45-x\left)\right)$

$⟹1+\tan x+\tan (45-x)+\tan x\tan (45-x)$

$⟹1+\tan (x+45-x)(1-\tan x\tan (45-x\left)\right)+\tan x\tan (45-x)$

$⟹2$

There are $22$ such terms in given sequence

$⟹\left(2^{22}\right)(1+\tan {45}^{\circ })$

$⟹2^{22}\times 2=2^{23}$