Write the value of the expression 1-4 10^ 70^ 2 10^ . - Vidyadip

Question

Write the value of the expression $\frac{1-4\sin10^\circ\sin70^\circ}{2\sin10^\circ}.$

✓

Answer

We have, $\frac{1-4\sin10^\circ\sin70^\circ}{2\sin10^\circ}$ $=\ \frac{1-2(2\sin10^\circ\sin70^\circ)}{2\sin10^\circ}$ $=\ \frac{1-2\big[\cos(70^\circ-10^\circ)-\cos(70^\circ+10^\circ)\big]}{2\sin10^\circ}$ $=\ \frac{1-2(\cos60^\circ-\cos80^\circ)}{2\sin(90^\circ-10^\circ)}$ $=\ \frac{1-2\big(\frac{1}{2}-\cos80^\circ\big)}{2\cos80^\circ}$ $=\ \frac{1-\frac{2}{2}+2\cos80^\circ}{2\cos80^\circ}$ $=\ \frac{1-1+2\cos80^\circ}{\cos80^\circ}$ $=\ \frac{2\cos80^\circ}{2\cos80^\circ}$ $=\ 1$

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