5.2.2 For what value(s) of \( x \) in the interval \( x \in\left[0^{\circ} ; 360^{\circ}\right] \) is \( \frac{\cos ^{4} x+\sin ^{2} x \cdot \cos ^{2} x}{1+\sin x} \) undefined?

To determine for which values of \( x \) in the interval \( [0^\circ, 360^\circ] \) the expression \[ \frac{\cos^4 x + \sin^2 x \cdot \cos^2 x}{1 + \sin x} \] is undefined, let's analyze the expression step-by-step. ### Step 1: Identify When the Expression is Undefined A rational expression is undefined when its denominator is zero (and the numerator is not also zero). Therefore, we need to solve: \[ 1 + \sin x = 0 \quad \Rightarrow \quad \sin x = -1 \] ### Step 2: Find \( x \) Values Where \( \sin x = -1 \) Within the interval \( [0^\circ, 360^\circ] \), \( \sin x = -1 \) at: \[ x = 270^\circ \] ### Step 3: Verify the Numerator at \( x = 270^\circ \) Evaluate the numerator at \( x = 270^\circ \): \[ \cos(270^\circ) = 0 \quad \text{and} \quad \sin(270^\circ) = -1 \] So, \[ \cos^4(270^\circ) + \sin^2(270^\circ) \cdot \cos^2(270^\circ) = 0 + (-1)^2 \cdot 0 = 0 \] This results in the form \( \frac{0}{0} \), which is undefined. ### Step 4: Simplify the Expression For additional clarity, let's simplify the original expression: \[ \frac{\cos^4 x + \sin^2 x \cdot \cos^2 x}{1 + \sin x} = \frac{\cos^2 x (\cos^2 x + \sin^2 x)}{1 + \sin x} = \frac{\cos^2 x}{1 + \sin x} \] Further simplification by multiplying the numerator and denominator by \( 1 - \sin x \): \[ \frac{\cos^2 x (1 - \sin x)}{1 - \sin^2 x} = \frac{\cos^2 x (1 - \sin x)}{\cos^2 x} = 1 - \sin x \quad (\text{provided } \cos x \neq 0) \] However, at \( x = 270^\circ \), both \( \cos x = 0 \) and \( 1 + \sin x = 0 \), leading back to the undefined form \( \frac{0}{0} \). ### Conclusion The expression is undefined only at: \[ x = 270^\circ \] **Answer:** The expression is undefined at 270°.