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

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UpStudy AI · Accepted Answer

Given the equation  
$$
9 \cos A - 7 = 0,
$$  
we start by isolating $$\cos A$$. Add $$7$$ to both sides and then divide by $$9$$:  
$$
\cos A = \frac{7}{9}.
$$

Next, recall the Pythagorean identity:  
$$
\sin^2 A + \cos^2 A = 1.
$$  
Substitute $$\cos A = \frac{7}{9}$$:  
$$
\sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{7}{9}\right)^2 = 1 - \frac{49}{81} = \frac{81 - 49}{81} = \frac{32}{81}.
$$

The cosecant squared is defined as the reciprocal of $$\sin^2 A$$:  
$$
\csc^2 A = \frac{1}{\sin^2 A} = \frac{1}{\frac{32}{81}} = \frac{81}{32}.
$$

Thus, the value of $$\csc^2 A$$ is  
$$
\boxed{\frac{81}{32}}.
$$
$$ \csc^2 A = \frac{81}{32} $$

If , find the value of .

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