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### Answers (1)

Answer:

$D=1$

Hint:

Determinant matrix must be square (i.e. same number of rows and columns)

Given:

\begin{aligned} \left|\begin{array}{cc} \sin 10^{\circ} & -\cos 10^{\circ} \\ \sin 80^{\circ} & \cos 80^{\circ} \end{array}\right|\\ \end{aligned}

Solution:

\begin{aligned} &\Delta=\mathrm{a}_{11} \mathrm{C}_{11}+\mathrm{a}_{21} \mathrm{C}_{21}\\ &=\sin 10^{\circ} \cos 80^{\circ}+\cos 10^{\circ} \sin 80^{\circ}\\ &=\sin 10^{\circ} \cos \left(90^{\circ}-10^{\circ}\right)+\cos 10^{\circ} \sin \left(90^{\circ}-10^{\circ}\right) \quad\quad\quad \quad\left[\sin \left(90^{\circ}-\theta\right)=\cos \theta\right]\\ &=\sin 10^{\circ} \sin 10^{\circ}+\cos 10^{\circ} \cos 10^{\circ}\\ &=\sin ^{2} 10^{\circ}+\cos ^{2} 10^{\circ} \quad \quad \quad \quad \quad\left[\cos ^{2} \theta+\sin ^{2} \theta=1\right]\\ &=1 \end{aligned}

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