--- ### Evaluating Expressions Using Trigonometric Function Values of Quadrantal Angles **Problem Statement:** Use the trigonometric function values of the quadrantal angles to evaluate the given expression: \[ 9(\sin 90^\circ)^2 + 4(\cot 90^\circ)^2 + \csc 270^\circ \] --- \[ 9(\sin 90^\circ)^2 + 4(\cot 90^\circ)^2 + \csc 270^\circ = \boxed{\phantom{10}} \] *(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)* --- In this exercise, you are tasked with simplifying a given trigonometric expression. You should use the exact values of the trigonometric functions at the specified quadrantal angles: \(90^\circ\) and \(270^\circ\). ### Step-by-Step Solution **1. Evaluate each trigonometric function component:** - **\(\sin 90^\circ\):** \[ \sin 90^\circ = 1 \] - **\(\cot 90^\circ\):** \[ \cot 90^\circ = \frac{\cos 90^\circ}{\sin 90^\circ} = \frac{0}{1} = 0 \] - **\(\csc 270^\circ\):** \[ \csc 270^\circ = \frac{1}{\sin 270^\circ} = \frac{1}{-1} = -1 \] **2. Substitute these values into the expression:** \[ 9(\sin 90^\circ)^2 = 9(1)^2 = 9 \] \[ 4(\cot 90^\circ)^2 = 4(0)^2 = 0 \] \[ \csc 270^\circ = -1 \] **3. Combine the results:** \[ 9 + 0 + (-1) = 8 \] ### Final Answer The simplified value of the expression is: \[ 8 \] --- By understanding and using the exact values of the trigonometric functions at quadrantal angles, you can correctly evaluate and simplify the given expression. Make sure to pay close attention to each step in order to arrive at the correct answer