Question

If the value of $(1+\cot {89}^{\circ })(1+\cot {88}^{\circ })(1+\cot {87}^{\circ })....(1+\cot {45}^{\circ })=p^q,p,q\in N$, then

• A. $p+q$ is irrational
• B. $|p-q|$ is a prime number
• C. $p+q$ is is a perfect square
• D. $q$ is integral multiple of $p$

Answer 1

The correct option is C $p+q$ is is a perfect square

$S=(1+\cot {89}^{\circ })(1+\cot {88}^{\circ })(1+\cot {87}^{\circ })....(1+\cot {45}^{\circ })=(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })....(1+\tan {45}^{\circ })$

Now,
$\tan {45}^{\circ }=\tan (x^{\circ }+(45-x{)}^{\circ })\Rightarrow 1=\frac{\tan x^{\circ }+\tan (45-x{)}^{\circ }}{1-\tan x^{\circ }\tan (45-x{)}^{\circ }}\Rightarrow (1+\tan x^{\circ })(1+\tan (45-x{)}^{\circ })=2$

$\Rightarrow (1+\tan 1^{\circ })(1+\tan {44}^{\circ })=(1+\tan 2^{\circ })(1+\tan {43}^{\circ })=(1+\tan 3^{\circ })(1+\tan {42}^{\circ })......=(1+\tan {22}^{\circ })(1+\tan {23}^{\circ })=2$

$\Rightarrow S=2^{23}p=2,q=23$