Determine the exact value of $\csc ^{2}\left(225^{\circ}\right)+\cos ^{2}\left(120^{\circ}\right)$ a) 4 b) $\frac{5}{4}$ c) $\frac{9}{4}$ d) $\frac{1}{4}$

Step 1 ：Given the expression is \(\csc ^{2}\left(225^{\circ}\right)+\cos ^{2}\left(120^{\circ}\right)\).

Step 2 ：Recall that the cosecant function is the reciprocal of the sine function. So, \(\csc ^{2}\left(225^{\circ}\right)\) is equivalent to \(1/\sin ^{2}\left(225^{\circ}\right)\).

Step 3 ：We need to calculate the sine of 225 degrees and the cosine of 120 degrees, square them, and add the results.

Step 4 ：\(\sin ^{2}\left(225^{\circ}\right)\) is approximately 0.5 and \(\cos ^{2}\left(120^{\circ}\right)\) is approximately 0.25.

Step 5 ：Adding these results gives 2.25.

Step 6 ：Final Answer: The exact value of \(\csc ^{2}\left(225^{\circ}\right)+\cos ^{2}\left(120^{\circ}\right)\) is \(\boxed{\frac{9}{4}}\).