Question

The value of:-
$\left(1+\cos \frac{\pi }{8}\right)\left(1+\cos \frac{3\pi }{8}\right)\left(1+\cos \frac{5\pi }{8}\right)\left(1+\cos \frac{7\pi }{8}\right)$ is

Answer 1

$(1+\cos \frac{\pi }{8})(1+\cos \frac{3\pi }{8})(1+\cos \frac{7\pi }{8})$

$=\cos (\pi -\theta )-\cos \theta$

$=(1+\cos \frac{\pi }{8})(1-\cos \frac{\pi }{8})(1+\cos \frac{3\pi }{8})(1+\cos \frac{5\pi }{8})$

$={\sin }^2\frac{\pi }{8}\times \frac{{\sin }^{2}3\pi }{8}$

Put the rule of $\sin \frac{\pi }{8}c\sin \frac{3\pi }{8}$

$=(0.3826{)}^2\times (0.923{)}^2$

$=0.1248$