Question:
The value of \(\cos\frac{5\pi}{17}+\cos\frac{7\pi}{17}+2\cos\frac{11\pi}{17}\cos\frac{\pi}{17}\) is
The value of \(\cos\frac{5\pi}{17}+\cos\frac{7\pi}{17}+2\cos\frac{11\pi}{17}\cos\frac{\pi}{17}\) is
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Use \(2\cos A\cos B=\cos(A+B)+\cos(A-B)\), then apply \(\cos(\pi-\theta)=-\cos\theta\).
Updated On: May 7, 2026
- \(0\)
- \(1\)
- \(-1\)
- \(\frac{1}{2}\)
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The Correct Option is A
Solution and Explanation
We need to simplify:
\[
\cos\frac{5\pi}{17}+\cos\frac{7\pi}{17}+2\cos\frac{11\pi}{17}\cos\frac{\pi}{17}.
\]
Use the identity:
\[
2\cos A\cos B=\cos(A+B)+\cos(A-B).
\]
Here,
\[
A=\frac{11\pi}{17},\quad B=\frac{\pi}{17}.
\]
So,
\[
2\cos\frac{11\pi}{17}\cos\frac{\pi}{17}
=
\cos\left(\frac{11\pi}{17}+\frac{\pi}{17}\right)
+
\cos\left(\frac{11\pi}{17}-\frac{\pi}{17}\right).
\]
\[
=
\cos\frac{12\pi}{17}
+
\cos\frac{10\pi}{17}.
\]
Therefore, the given expression becomes:
\[
\cos\frac{5\pi}{17}
+\cos\frac{7\pi}{17}
+\cos\frac{12\pi}{17}
+\cos\frac{10\pi}{17}.
\]
Now use:
\[
\cos(\pi-\theta)=-\cos\theta.
\]
Observe:
\[
\frac{12\pi}{17}=\pi-\frac{5\pi}{17}.
\]
So,
\[
\cos\frac{12\pi}{17}
=
-\cos\frac{5\pi}{17}.
\]
Also,
\[
\frac{10\pi}{17}=\pi-\frac{7\pi}{17}.
\]
So,
\[
\cos\frac{10\pi}{17}
=
-\cos\frac{7\pi}{17}.
\]
Substitute:
\[
\cos\frac{5\pi}{17}
+\cos\frac{7\pi}{17}
-\cos\frac{5\pi}{17}
-\cos\frac{7\pi}{17}.
\]
All terms cancel:
\[
0.
\]
Hence, the value of the expression is
\[
0.
\]
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