Compute $\int\limits_0^{\pi}\ln\left(1-\cos\left(t\right)\right)\ dt$

Question

I want to show that $$ \int\limits_0^\pi \ln\left(1-\cos\left(t\right)\right)\text{d}t=-\pi\ln\left(2\right) $$ I wanted to use the integral $\displaystyle \int\limits_0^\pi \ln\sin\left(t\right)\ dt$ which I know the value but I struggle finding a judicious change of variable. Is there a way to do it ?

Answer 1

Hint:

As $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx=I$(say),

$$I+I=\int_a^bf(x)+f(a+b-x)\ dx$$

Now use Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$