Question

Show that

\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]

Answer 1

\[\text{Let} ∆ = \begin{vmatrix}\sin10^\circ & - \cos10^\circ \\ \sin80^\circ & \cos80^\circ\end{vmatrix}\]
\[ \Rightarrow ∆ = \sin10^\circ\cos80^\circ + \cos10^\circ\sin80^\circ\]
\[ = \sin10^\circ\cos(90^\circ - 10^\circ) + \cos10^\circ\sin(90^\circ - 10^\circ) \left[ \because \cos\theta = \sin\left( 90 - \theta \right) \right]\]
\[ \Rightarrow ∆ = \sin10^\circ\sin10^\circ + \cos10^\circ\cos10^\circ\]
\[ = \sin^2 10^\circ + \cos^2 10^\circ \left[ \because \sin^2 \theta + co s^2 \theta = 1 \right]\]
\[ \Rightarrow ∆ = 1\]