$16\left(\cos \mathrm{\theta }-\cos \frac{\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{3\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{5\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{7\mathrm{\pi }}{8}\right)=\mathrm{\lambda cos}4\mathrm{\theta }$ then the value of $\mathrm{\lambda }$ is

$16\left(\cos \mathrm{\theta }-\cos \frac{\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{3\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{5\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{7\mathrm{\pi }}{8}\right)$

$=16\left(\cos \mathrm{\theta }-\cos \frac{\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{7\mathrm{\pi }}{8}\right)\times \left(\cos \mathrm{\theta }-\cos \frac{3\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }-\cos \frac{5\mathrm{\pi }}{8}\right)$

$=16\left(\cos \mathrm{\theta }-\cos \frac{\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }+\cos \frac{\mathrm{\pi }}{8}\right)\times \left(\cos \mathrm{\theta }-\cos \frac{3\mathrm{\pi }}{8}\right)\left(\cos \mathrm{\theta }+\cos \frac{3\mathrm{\pi }}{8}\right)$

$\begin{array}{c}=16\left({\cos }^2\mathrm{\theta }-{\cos }^2\frac{\mathrm{\pi }}{8}\right)\left({\cos }^2\mathrm{\theta }-{\cos }^2\frac{3\mathrm{\pi }}{8}\right)\\ =16\left({\cos }^2\mathrm{\theta }-{\cos }^2\frac{\mathrm{\pi }}{8}\right)\left({\cos }^2\mathrm{\theta }-{\sin }^2\frac{\mathrm{\pi }}{8}\right)\\ =16\left({\cos }^4\mathrm{\theta }-{\cos }^2\mathrm{\theta }+{\sin }^2\frac{\mathrm{\pi }}{8}{\cos }^2\frac{\mathrm{\pi }}{8}\right)\\ =16\left({\cos }^4\mathrm{\theta }-{\cos }^2\mathrm{\theta }+\frac{1}{8}\right)\\ =16\left(-{\cos }^2{\mathrm{\theta sin}}^2\mathrm{\theta }+\frac{1}{8}\right)\\ =16\left(\frac{-{\sin }^{2}2\mathrm{\theta }}{4}+\frac{1}{8}\right)\\ =16\left(\frac{1-2{\sin }^{2}2\mathrm{\theta }}{8}\right)\\ =\frac{16\cos 4\mathrm{\theta }}{8}\end{array}$