The value of \( 2\cos\frac{\pi}{13}\cos\frac{9\pi}{13}+\cos\frac{3\pi}{13}+\cos\frac{5\pi}{13} \) is
Solution:
Using identity,
\[
2\cos A\cos B
=
\cos(A+B)+\cos(A-B)
\]
\[
2\cos\frac{\pi}{13}\cos\frac{9\pi}{13}
=
\cos\frac{10\pi}{13}+\cos\frac{8\pi}{13}
\]
Therefore,
\[
=
\cos\frac{10\pi}{13}
+
\cos\frac{8\pi}{13}
+
\cos\frac{3\pi}{13}
+
\cos\frac{5\pi}{13}
\]
Using,
\[
\cos(\pi-\theta)=-\cos\theta
\]
\[
\cos\frac{10\pi}{13}
=
-\cos\frac{3\pi}{13}
\]
and
\[
\cos\frac{8\pi}{13}
=
-\cos\frac{5\pi}{13}
\]
Hence,
\[
=
-\cos\frac{3\pi}{13}
-\cos\frac{5\pi}{13}
+\cos\frac{3\pi}{13}
+\cos\frac{5\pi}{13}
\]
\[
=0
\]
\[
\boxed{0}
\]