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  4. The value of (1+ cos (π/6))(1+ cos (π/3))(1+ cos (2 π/3)) (1+ cos (7 π/6)) is

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Q. The value of $\left(1+\cos \frac{\pi}{6}\right)\left(1+\cos \frac{\pi}{3}\right)\left(1+\cos \frac{2 \pi}{3}\right)$ $\left(1+\cos \frac{7 \pi}{6}\right)$ is

Trigonometric Functions

Solution:

$(1+ \left.\cos \frac{\pi}{6}\right)\left(1+\cos \frac{\pi}{3}\right)\left(1+\cos \frac{2 \pi}{3}\right)\left(1+\cos \frac{7 \pi}{6}\right) $
$=\left(1+\frac{\sqrt{3}}{2}\right)\left(1+\frac{1}{2}\right)\left(1-\frac{1}{2}\right)\left(1-\frac{\sqrt{3}}{2}\right)$
$ =\left(1-\frac{3}{4}\right)\left(1-\frac{1}{4}\right) $
$ =\frac{1}{4} \times \frac{3}{4}=\frac{3}{16}$