`T h ev a l u eof(lim)_(nvecoo)[t a npi/(2n)tan(2pi)/(2n)dottan(npi)/(2n)]^(1//n)i s` `e` (b) `e^2` (c) 1 (d) `e^3`

Options

A. `e`
B. `e^(2)`
C. `1`
D. `e^(3)`

Answer

C

Explanation

Let `A=lim_(n to oo) ["tan"(pi)/(2n)"tan"(2pi)/(2n)……"tan"(npi)/(2n)]^(1//n)`
`=:. logA=lim_(n to oo )1/n[log "tan"(pi)/(2n)+"log tan"(2pi)/(2n)+……..+"log tan"(npi)/(2n)]`
`=lim_(n to oo) sum_(r=1)^(n)(1/n) "log tan"(pir)/(2n)`
`=int_(0)^(1)"log tan ((pi)/2 x)dx`
`=2/(pi) int_(0)^(pi//2) log tan y dy `...............1
[ Putting `1/2 pi x=y` and `dx=(2//pi)dy`]
Now let `I=int_(0)^(pi//2) log tan y dy`
`:. I=int_(0)^(pi//2) log tan (1/2 pi -y) dy`
`=int_(0)^(pi//2) log cot y dy`
`=-int_(0)^(pi//2) log tan y dy=-1`
or `I+I=0` or `2I=0` or `I=0`
Thus, from equation 1 `log A=0` or `A=e^(0)=1`