If \(\sin 6A = \cos 12A\), find the value of \(\tan 9A + \cot 9A\).
2
Use the complementary identity \(\sin\theta = \cos(90^\circ - \theta)\), so \(\sin 6A = \cos(90^\circ - 6A)\).
Equating with \(\cos 12A\): \(90^\circ - 6A = 12A\).
Solving: \(90^\circ = 18A\), so \(A = 5^\circ\).
Then \(9A = 45^\circ\), and \(\tan 45^\circ = 1\), \(\cot 45^\circ = 1\).
Therefore \(\tan 9A + \cot 9A = 1 + 1 = 2\).
Hence, the value of \(\tan 9A + \cot 9A\) is 2.
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