Refer to this for part a: https://math.stackexchange.com/questions/1760925/lim-n-to-infty-fracn1n2-dots3nn2n-frac1n-is-equa/1760934

lim n→∞ { (1 + 1/n)(1 + 2/n).....(1 + n/n) }¹/ⁿ

= lim n→∞ e^ [ln{ (1 + 1/n)(1 + 2/n).....(1 + n/n) }¹/ⁿ]

= lim n→∞ e^ [ 1/n ln{ (1 + 1/n)(1 + 2/n).....(1 + n/n) } ]

= e^(lim n→∞ 1/n { ln(1 + 1/n) + ln(1 + 2/n)+ ln(1 + n/n) })

= e^ ∫ ln(x + 1) dx (upper limit 1, lower 0)

= e^ [(x + 1)ln(x + 1) - x] (upper limit 1, lower 0)

= e^(2ln2 - 1)

= e^(ln2² - 1)

= e^(ln4) 1/e

= 4/e

Part ii seems tough. Apparently the question requires a concept that is either beyond my ability or I have not touched for years: to convert a power n to an angle multiple n.

I obtained a different integral (m + n - 1) / mn times integral (0 to pi/2) of [cos^(m + n - 2) x].

It requires the use of a more advanced power formula for cos²ⁿ+¹ x and cos²ⁿ x

The qn is abit wrong apparently. Is cosnx not cos^n(x)

The question looks valid (indeed, in the demoivre’s formula, it is possible to bring a power down). But I am not sure of how to invoke it here.

If it's cos nx instead of cosⁿx then it's easy already