If tanα + sinβ = m

cotα + cosecβ = n, α ≠ β, then

find the value of cosec²β (\(\frac{m}{n}\))² + 4.

sec²α + cot²β

Put α = 45°, β = 30°

[∵ Two equations and 4 variables are given, put 2 of them any thing]

⇒ tanα + sinβ = m

m = 1 + \(\frac{1}{2}\) = \(\frac{3}{2}\)

⇒ cotα + cosecβ = n

n = 1 + 2 = 3

Put in find out

⇒ cosec²β (\(\frac{m}{n}\))² + 4

⇒ (2)² × \(\frac{1}{4}\) + 4 = 5

Now check options and satisfy.

From option A:

⇒ sec²α + cot²β = (\(\sqrt {2}\))² + (\(\sqrt {3}\))² = 5 satisfied

⇒ sec²α + cot²β is the correct answer.