If x and y are positive acute angles such that sin(4x – y) = 1 and cos(2x + y) = $frac{1}{2}$,, then what is the value of cot (x + 2y) ?

sin (4x – y) = 1 = sin 90° ⇒ 4x – y = 90° ..... (i) Again, $cos (2 x+y)=frac{1}{2}=cos 60^{circ}$ ⇒ 2x + y = 60° .... (ii) By equations (i) + (ii),

$Rightarrow x=frac{150^{circ}}{6}=25^{circ}$

From equation (i)

4 × 25° – y = 90°

⇒ y = 100° – 90° = 10°

:. cot (x + 2y) = cot (25° + 20°) = cot 45° = 1