If $$\theta \in[0,2 \pi]$$ and $$\cos 2 \theta=\cos \theta+\sin \theta$$, then the sum of all values of $$\theta$$ satisfying the equation is

For how many distinct values of $$x$$, the following $$\sin \left[2 \cos ^{-1} \cot \left(2 \tan ^{-1} x\right)\right]=0$$ holds?

In $$\triangle A B C$$, suppose the radius of the circle opposite to an angle $$A$$ is denoted by $$r_1$$, similarly $$r_2 \leftrightarrow$$ angle $$B, r_3 \leftrightarrow$$ angle $$C$$. If $$r_1=2, r_2=3, r_3=6$$, what is the value of $$r_1+r_2+r_3-r=$$ (R - radius of the circum circle).

In $$\triangle A B C \cdot \frac{a+b+c}{B C+A B}+\frac{a+b+c}{A C+A B}=3$$, then $$\tan \frac{C}{8}$$ is equal to