Question:

<p><strong>Step 1:</strong> From \(\sec x + \tan x = \frac{22}{7}\), use the identity \((\sec x + \tan x)(\sec x - \tan x) = 1\).</p><p><strong>Step 2:</strong> Therefore \(\sec x - \tan x = \frac{7}{22}\).</p><p><strong>Step 3:</strong> Adding: \(2\sec x = \frac{22}{7} + \frac{7}{22} = \frac{484 + 49}{154} = \frac{533}{154}\), so \(\sec x = \frac{533}{308}\) and \(\cos x = \frac{308}{533}\).</p><p><strong>Step 4:</strong> Subtracting: \(2\tan x = \frac{22}{7} - \frac{7}{22} = \frac{484 - 49}{154} = \frac{435}{154}\), so \(\tan x = \frac{435}{308}\).</p><p><strong>Step 5:</strong> Find \(\sin x = \tan x \cdot \cos x = \frac{435}{308} \cdot \frac{308}{533} = \frac{435}{533}\).</p><p><strong>Step 6:</strong> Calculate \(\cosec x + \cot x = \frac{1}{\sin x} + \frac{\cos x}{\sin x} = \frac{1 + \cos x}{\sin x} = \frac{1 + 308/533}{435/533} = \frac{841/533}{435/533} = \frac{841}{435} = \frac{29}{15}\).</p><p>∴ Answer is C.</p>