Show that 10^ &- 10^ 80^ & 80^ =1 - Vidyadip

Question

Show that $\begin{vmatrix}\sin10^\circ&-\cos10^\circ\\\sin80^\circ&\cos80^\circ \end{vmatrix}=1$

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Answer

Let $\triangle=\begin{vmatrix}\sin10^\circ&-\cos10^\circ\\\sin80^\circ&\cos80^\circ \end{vmatrix}$
$\Rightarrow\triangle=\sin10^\circ\cos80^\circ+\cos10^\circ\sin80^\circ$
$=\sin10^\circ\cos(90^\circ-10^\circ)+\cos10^\circ\sin(90^\circ-10^\circ)$ $\big[\because\cos\theta=\sin(90-\theta)\big]$
$\Rightarrow\triangle=\sin10^\circ\sin10^\circ+\cos10^\circ\cos10^\circ$
$=\sin^210^\circ+\cos^210^\circ$
$\Rightarrow\triangle=1$ $\big[\because\sin^2\theta+\cos^2\theta=1\big]$

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