For \( A = \frac{\pi}{24} \), if \( \frac{\sin 2A + \sin 3A + \sin 4A}{\cos 2A + \cos 3A + \cos 4A} = k \), then \( (k+1)^2 = \)
Show Hint
To simplify expressions like \( \frac{\sin nx + \sin (n+1)x + \sin (n+2)x}{\cos nx + \cos (n+1)x + \cos (n+2)x} \), the result is always \( \tan((n+1)x) \). This "middle angle" property is a common shortcut for symmetric sums.