If \( 9 \cos A-7=0 \), find the value of \( \operatorname{cosec}^{2} A \).

To solve the equation \( 9 \cos A - 7 = 0 \), we first isolate \( \cos A \): \[ 9 \cos A = 7 \] \[ \cos A = \frac{7}{9} \] Next, we need to find \( \operatorname{cosec}^{2} A \). We can use the identity relating cosecant and cosine: \[ \csc^2 A = \frac{1}{\sin^2 A} \] To find \( \sin^2 A \), we can use the Pythagorean identity: \[ \sin^2 A + \cos^2 A = 1 \] Substituting \( \cos A = \frac{7}{9} \): \[ \sin^2 A + \left(\frac{7}{9}\right)^2 = 1 \] Calculating \( \left(\frac{7}{9}\right)^2 \): \[ \sin^2 A + \frac{49}{81} = 1 \] Now, subtract \( \frac{49}{81} \) from both sides: \[ \sin^2 A = 1 - \frac{49}{81} \] To express 1 with a common denominator: \[ 1 = \frac{81}{81} \] Thus: \[ \sin^2 A = \frac{81}{81} - \frac{49}{81} = \frac{32}{81} \] Now, we substitute \( \sin^2 A \) into the cosecant formula: \[ \csc^2 A = \frac{1}{\sin^2 A} = \frac{1}{\frac{32}{81}} = \frac{81}{32} \] Therefore, the value of \( \operatorname{cosec}^{2} A \) is \[ \boxed{\frac{81}{32}} \]