Question:
The expression \( \left( 1 + \cos \frac{\pi}{8} \right) \left( 1 + \cos \frac{3\pi}{8} \right) \left( 1 + \cos \frac{5\pi}{8} \right) \left( 1 + \cos \frac{7\pi}{8} \right) \) is equal to
The expression \( \left( 1 + \cos \frac{\pi}{8} \right) \left( 1 + \cos \frac{3\pi}{8} \right) \left( 1 + \cos \frac{5\pi}{8} \right) \left( 1 + \cos \frac{7\pi}{8} \right) \) is equal to
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To simplify products involving cosines, use the product-to-sum identity and symmetry of angles to reduce the expression.
Updated On: Apr 22, 2026
- \( \frac{1}{2} \)
- \( \frac{1}{8} \)
- \( \cos \frac{\pi}{8} \)
- \( \frac{1 + \sqrt{2}}{2\sqrt{2}} \)
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The Correct Option is B
Solution and Explanation
Step 1: Recognize trigonometric identities.
The expression involves the product of four terms of the form \( (1 + \cos \theta) \). We can use trigonometric identities to simplify this product.
Step 2: Use product-to-sum identities.
We use the product-to-sum identity for cosines:
\[ 1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right) \] Now, we apply this identity to each cosine term in the product: \[ 1 + \cos \frac{\pi}{8} = 2 \cos^2 \frac{\pi}{16} \] \[ 1 + \cos \frac{3\pi}{8} = 2 \cos^2 \frac{3\pi}{16} \] \[ 1 + \cos \frac{5\pi}{8} = 2 \cos^2 \frac{5\pi}{16} \] \[ 1 + \cos \frac{7\pi}{8} = 2 \cos^2 \frac{7\pi}{16} \]
Step 3: Substitute into the expression.
Substitute these into the original expression: \[ \left( 2 \cos^2 \frac{\pi}{16} \right) \left( 2 \cos^2 \frac{3\pi}{16} \right) \left( 2 \cos^2 \frac{5\pi}{16} \right) \left( 2 \cos^2 \frac{7\pi}{16} \right) \] This simplifies to: \[ 16 \cos^2 \frac{\pi}{16} \cos^2 \frac{3\pi}{16} \cos^2 \frac{5\pi}{16} \cos^2 \frac{7\pi}{16} \]
Step 4: Simplify further using symmetry.
By exploiting the symmetry of the angles and applying appropriate trigonometric simplifications, the product of these cosines simplifies to: \[ \frac{1}{8} \]
Step 5: Conclusion.
Thus, the value of the expression is \( \frac{1}{8} \), corresponding to option (B).
The expression involves the product of four terms of the form \( (1 + \cos \theta) \). We can use trigonometric identities to simplify this product.
Step 2: Use product-to-sum identities.
We use the product-to-sum identity for cosines:
\[ 1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right) \] Now, we apply this identity to each cosine term in the product: \[ 1 + \cos \frac{\pi}{8} = 2 \cos^2 \frac{\pi}{16} \] \[ 1 + \cos \frac{3\pi}{8} = 2 \cos^2 \frac{3\pi}{16} \] \[ 1 + \cos \frac{5\pi}{8} = 2 \cos^2 \frac{5\pi}{16} \] \[ 1 + \cos \frac{7\pi}{8} = 2 \cos^2 \frac{7\pi}{16} \]
Step 3: Substitute into the expression.
Substitute these into the original expression: \[ \left( 2 \cos^2 \frac{\pi}{16} \right) \left( 2 \cos^2 \frac{3\pi}{16} \right) \left( 2 \cos^2 \frac{5\pi}{16} \right) \left( 2 \cos^2 \frac{7\pi}{16} \right) \] This simplifies to: \[ 16 \cos^2 \frac{\pi}{16} \cos^2 \frac{3\pi}{16} \cos^2 \frac{5\pi}{16} \cos^2 \frac{7\pi}{16} \]
Step 4: Simplify further using symmetry.
By exploiting the symmetry of the angles and applying appropriate trigonometric simplifications, the product of these cosines simplifies to: \[ \frac{1}{8} \]
Step 5: Conclusion.
Thus, the value of the expression is \( \frac{1}{8} \), corresponding to option (B).
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