The value of lim n→∞ ​[tan π/2n . tan 2π/2n .....tan nπ/2n]^1/n

Question

The value of \(\lim\limits_{n \to \infty} \left[\tan \frac\pi {2n}.\tan\frac{2\pi}{2n}...\tan \frac{n\pi}{2n}\right]^{1/n}\)

(a) e

(b) e²

(c) 1

(d) e³ 

Answer 1

Correct option is (c) 1

(by Property IV)

= ∫₀^π/2 ​log coty dy

= −∫₀^π/2​l og tany dy = −I​

or I + I = 0 

or 2I = 0

or I = 0

logA = 0

∴ A = e⁰ = 1