\( \begin{array}{ll}5.5 .1 & \text { Rewrite the expression as a single trigonometric ratio. } \\ 5.5 .2 & \text { For which value of } x \text { in the interval } x \in\left[0^{\circ} ; 90^{\circ}\right] \text { will } \\ & 16 \sin x \cdot \cos ^{3} x-8 \sin x \cdot \cos x \text { have its minimum value? }\end{array} \)

To rewrite the expression \( 16 \sin x \cdot \cos^3 x - 8 \sin x \cdot \cos x \) as a single trigonometric ratio, you can factor out the common term, \( 8 \sin x \cdot \cos x \). This gives us: \[ 8 \sin x \cdot \cos x (2 \cos^2 x - 1) \] Using the double angle identity, \( \sin 2x = 2 \sin x \cdot \cos x \), you can convert this into: \[ 4 \sin 2x (2 \cos^2 x - 1) \] For the second part, to find the value of \( x \) that minimizes \( 16 \sin x \cdot \cos^3 x - 8 \sin x \cdot \cos x \), we observe that this expression is proportional to \( \sin x \cdot \cos x (2\cos^2 x - 1) \). The function \( 2\cos^2 x - 1 = \cos 2x \), hence we are looking for when this expression reaches its minimum—typically at values where \( \sin x \) is minimized while remaining non-negative in the interval \( [0^\circ, 90^\circ] \). Evaluating, we find that at \( x = 90^\circ \), \( \sin x = 1 \), and the cosine terms yield zero, thus achieving a minimum. Therefore, \( x = 90^\circ \) gives the minimum value of the original expression.