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Question
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
Solution
\[\frac{1 - 4\sin10^\circ \sin70^\circ}{2\sin10^\circ}\]
\[ = \frac{1 - 2\left[ 2\sin10^\circ \sin70^\circ \right]}{2\sin10^\circ}\]
\[ = \frac{1 - 2\left[ \cos\left( 10^\circ - 70^\circ \right) - \cos\left( 10^\circ + 70^\circ \right) \right]}{2\sin10^\circ}\]
\[ = \frac{1 - 2\left[ \cos\left( - 60^\circ \right) - \cos80^\circ \right]}{2\sin10^\circ}\]
\[ = \frac{1 - 2\left[ \cos60^\circ - \cos80^\circ \right]}{2\sin10^\circ}\]
\[ = \frac{1 - 2\left[ \frac{1}{2} - \cos\left( 90^\circ - 10^\circ \right) \right]}{2\sin10^\circ}\]
\[ = \frac{1 - 2 \times \frac{1}{2} + 2\cos\left( 90^\circ - 10^\circ \right)}{2\sin10^\circ}\]
\[ = \frac{2\sin10^\circ}{2\sin10^\circ}\]
\[ = 1\]
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