The sum of series $\log \cos \left(\frac{x}{2}\right)+\log \left(\cos \frac{x}{2^2}\right)+\log \lef

Question

The sum of series $\log \cos \left(\frac{x}{2}\right)+\log \left(\cos \frac{x}{2^2}\right)+\log \left(\cos \frac{x}{2^3}+\ldots \ldots \ldots+\log \left(\cos \frac{x}{2^n}\right)+\log \left(\sin \frac{x}{2^n}\right)\right)$ is equal to

Professor PC Thomas Classes · Answer

Given expression is $ \log \cos \frac{x}{2}+\log \cos \frac{x}{2^2}+\ldots+\log \cos \frac{x}{2^n}+\log \sin \frac{x}{2^n}+\log 2^n-\log 2^n $ by adding and subtracting $\log 2^n$ $ \log \sin x-\log 2^n=\log \left(\frac{\sin x}{2^n}\right) $