Write the value of \( \dfrac{1-4\sin10^\circ\sin70^\circ}{2\sin10^\circ} \)
Solution:
Using identity,
\[
2\sin A\sin B
=
\cos(A-B)-\cos(A+B)
\]
\[
2\sin10^\circ\sin70^\circ
=
\cos60^\circ-\cos80^\circ
\]
\[
=
\frac12-\cos80^\circ
\]
Multiplying by 2,
\[
4\sin10^\circ\sin70^\circ
=
1-2\cos80^\circ
\]
Therefore,
\[
1-4\sin10^\circ\sin70^\circ
=
2\cos80^\circ
\]
Hence,
\[
\frac{1-4\sin10^\circ\sin70^\circ}{2\sin10^\circ}
=
\frac{2\cos80^\circ}{2\sin10^\circ}
\]
Using,
\[
\cos80^\circ=\sin10^\circ
\]
\[
=
\frac{2\sin10^\circ}{2\sin10^\circ}
\]
\[
=1
\]
\[
\boxed{1}
\]