If \(\sin A = \frac{5}{7}\), find the value of \(\dfrac{5\,\text{cosec}\,A - 7\sin A}{2\sqrt{6}\,\sec A + 7\cos^2 A}\).
\(\frac{14}{73}\)
Given \(\sin A = \frac{5}{7}\), we have \(\cos A = \sqrt{1 - \frac{25}{49}} = \sqrt{\frac{24}{49}} = \frac{2\sqrt{6}}{7}\).
Then \(\text{cosec}\,A = \frac{7}{5}\), \(\sec A = \frac{7}{2\sqrt{6}}\) and \(\cos^2 A = \frac{24}{49}\).
Numerator: \(5\,\text{cosec}\,A - 7\sin A = 5 \times \frac{7}{5} - 7 \times \frac{5}{7} = 7 - 5 = 2\).
First denominator term: \(2\sqrt{6}\,\sec A = 2\sqrt{6} \times \frac{7}{2\sqrt{6}} = 7\).
Second denominator term: \(7\cos^2 A = 7 \times \frac{24}{49} = \frac{24}{7}\).
Denominator: \(7 + \frac{24}{7} = \frac{49 + 24}{7} = \frac{73}{7}\).
The expression is \(\dfrac{2}{\frac{73}{7}} = 2 \times \frac{7}{73} = \frac{14}{73}\).
Hence, the value of the expression is \(\frac{14}{73}\).
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