Q78 / 180 · COMEDK 2024 (Morning Shift) · Mathematics
$4\left(1+\cos \dfrac{\pi}{8}\right)\left(1+\cos \dfrac{3 \pi}{8}\right)\left(1+\cos \dfrac{5 \pi}{8}\right)\left(1+\cos \dfrac{7 \pi}{8}\right) \text { is equal to }$
- A1
- B$\dfrac{1}{8}$
- C$\dfrac{1}{4}$
- Correct answerD$\dfrac{1}{2}$
Solution
Let the given expression be $E = 4 \left(1 + \cos \dfrac{\pi}{8}\right) \left(1 + \cos \dfrac{3\pi}{8}\right) \left(1 + \cos \dfrac{5\pi}{8}\right) \left(1 + \cos \dfrac{7\pi}{8}\right)$.
Using the identity $\cos(\pi - \theta) = -\cos \theta$, we have $\cos \dfrac{7\pi}{8} = -\cos \dfrac{\pi}{8}$ and $\cos \dfrac{5\pi}{8} = -\cos \dfrac{3\pi}{8}$.
Substituting these into the expression:
$E = 4 \left(1 + \cos \dfrac{\pi}{8}\right) \left(1 + \cos \dfrac{3\pi}{8}\right) \left(1 - \cos \dfrac{3\pi}{8}\right) \left(1 - \cos \dfrac{\pi}{8}\right)$
Grouping the terms:
$E = 4 \left(1 - \cos^2 \dfrac{\pi}{8}\right) \left(1 - \cos^2 \dfrac{3\pi}{8}\right)$
$E = 4 \sin^2 \dfrac{\pi}{8} \sin^2 \dfrac{3\pi}{8}$
Using the identity $\sin^2 \theta = \dfrac{1 - \cos 2\theta}{2}$:
$E = 4 \left(\dfrac{1 - \cos(\pi/4)}{2}\right) \left(\dfrac{1 - \cos(3\pi/4)}{2}\right)$
$E = 4 \left(\dfrac{1 - 1/\sqrt{2}}{2}\right) \left(\dfrac{1 - (-1/\sqrt{2})}{2}\right)$
$E = 4 \left(\dfrac{1 - 1/\sqrt{2}}{2}\right) \left(\dfrac{1 + 1/\sqrt{2}}{2}\right)$
$E = 4 \left(\dfrac{1 - 1/2}{4}\right) = 4 \left(\dfrac{1/2}{4}\right) = 4 \times \dfrac{1}{8} = \dfrac{1}{2}$.
Answer: $\dfrac{1}{2}$
Using the identity $\cos(\pi - \theta) = -\cos \theta$, we have $\cos \dfrac{7\pi}{8} = -\cos \dfrac{\pi}{8}$ and $\cos \dfrac{5\pi}{8} = -\cos \dfrac{3\pi}{8}$.
Substituting these into the expression:
$E = 4 \left(1 + \cos \dfrac{\pi}{8}\right) \left(1 + \cos \dfrac{3\pi}{8}\right) \left(1 - \cos \dfrac{3\pi}{8}\right) \left(1 - \cos \dfrac{\pi}{8}\right)$
Grouping the terms:
$E = 4 \left(1 - \cos^2 \dfrac{\pi}{8}\right) \left(1 - \cos^2 \dfrac{3\pi}{8}\right)$
$E = 4 \sin^2 \dfrac{\pi}{8} \sin^2 \dfrac{3\pi}{8}$
Using the identity $\sin^2 \theta = \dfrac{1 - \cos 2\theta}{2}$:
$E = 4 \left(\dfrac{1 - \cos(\pi/4)}{2}\right) \left(\dfrac{1 - \cos(3\pi/4)}{2}\right)$
$E = 4 \left(\dfrac{1 - 1/\sqrt{2}}{2}\right) \left(\dfrac{1 - (-1/\sqrt{2})}{2}\right)$
$E = 4 \left(\dfrac{1 - 1/\sqrt{2}}{2}\right) \left(\dfrac{1 + 1/\sqrt{2}}{2}\right)$
$E = 4 \left(\dfrac{1 - 1/2}{4}\right) = 4 \left(\dfrac{1/2}{4}\right) = 4 \times \dfrac{1}{8} = \dfrac{1}{2}$.
Answer: $\dfrac{1}{2}$
Hence, the correct answer is option: 'D'.