(1)

\begin{align*} \sin A+\sin B+\sin C & =\sin A+\sin B+\sin C-\sin\left(A+B+C\right)+\sin\left(A+B+C\right)\\ & =2\sin\frac{A+B}{2}\cos\frac{A-B}{2}+\sin\left(-\frac{A+B}{2}\right)\cos\frac{A+B+2C}{2}+\sin\left(A+B+C\right)\\ & =2\sin\frac{A+B}{2}\left(\cos\frac{A-B}{2}-\cos\frac{A+B+2C}{2}\right)+\sin\left(A+B+C\right)\\ & =2\sin\frac{A+B}{2}\left\{ -2\sin\frac{A+C}{2}\sin\left(-\frac{B+C}{2}\right)\right\} +\sin\left(A+B+C\right)\\ & =4\sin\frac{B+C}{2}\sin\frac{C+A}{2}\sin\frac{A+B}{2}+\sin\left(A+B+C\right) \end{align*}

(2)

\begin{align*} \sin A\sin B\sin C & =-\frac{1}{2}\left\{ \cos\left(A+B\right)-\cos\left(A-B\right)\right\} \sin C\\ & =-\frac{1}{2}\left\{ \frac{1}{2}\left\{ \sin\left(A+B+C\right)+\sin\left(C-A-B\right)\right\} -\frac{1}{2}\left\{ \sin\left(A-B+C\right)+\sin\left(C-A+B\right)\right\} \right\} \\ & =\frac{1}{4}\left\{ \sin\left(A+B+C\right)+\sin\left(-A+B+C\right)+\sin\left(A-B+C\right)+\sin\left(A+B-C\right)\right\} \end{align*}

(3)

\begin{align*} \cos A+\cos B+\cos C & =\cos A+\cos B+\cos C+\cos\left(A+B+C\right)-\cos\left(A+B+C\right)\\ & =2\cos\frac{A+B}{2}\cos\frac{A-B}{2}+2\cos\frac{A+B+2C}{2}\cos\left(-\frac{A+B}{2}\right)-\cos\left(A+B+C\right)\\ & =2\cos\frac{A+B}{2}\left(\cos\frac{A-B}{2}+\cos\frac{A+B+2C}{2}\right)-\cos\left(A+B+C\right)\\ & =2\cos\frac{A+B}{2}\left(2\cos\frac{A+C}{2}\cos\left(-\frac{B+C}{2}\right)\right)-\cos\left(A+B+C\right)\\ & =4\cos\frac{B+C}{2}\cos\frac{C+A}{2}\cos\frac{A+B}{2}-\cos\left(A+B+C\right) \end{align*}

(4)

\begin{align*} \cos A\cos B\cos C & =\frac{1}{2}\left\{ \cos\left(A+B\right)+\cos\left(A-B\right)\right\} \cos C\\ & =\frac{1}{2}\left\{ \frac{1}{2}\left\{ \cos\left(A+B+C\right)+\cos\left(A+B-C\right)\right\} +\frac{1}{2}\left\{ \cos\left(A-B+C\right)+\cos\left(A-B-C\right)\right\} \right\} \\ & =\frac{1}{4}\left\{ \cos\left(A+B+C\right)+\cos\left(-A+B+C\right)+\cos\left(A-B+C\right)+\cos\left(A+B-C\right)\right\} \end{align*}

(5)

\begin{align*} \tan\left(A+B+C\right) & =\frac{\tan\left(A+B\right)+\tan C}{1-\tan\left(A+B\right)\tan C}\\ & =\frac{\frac{\tan A+\tan B}{1-\tan A\tan B}+\tan C}{1-\frac{\tan A+\tan B}{1-\tan A\tan B}\tan C}\\ & =\frac{\frac{\tan A+\tan B}{1-\tan A\tan B}+\tan C}{1-\frac{\tan A+\tan B}{1-\tan A\tan B}\tan C}\\ & =\frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan C\tan A} \end{align*}

(6)

\begin{align*} \tan^{-1}\left(A+B+C\right) & =\left(\frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan C\tan A}\right)^{-1}\\ & =\frac{1-\tan A\tan B-\tan B\tan C-\tan C\tan A}{\tan A+\tan B+\tan C-\tan A\tan B\tan C}\\ & =\frac{\tan^{-1}A\tan^{-1}B\tan^{-1}C-\tan^{-1}C-\tan^{-1}A-\tan^{-1}B}{\tan^{-1}B\tan^{-1}C+\tan^{-1}C\tan^{-1}A+\tan^{-1}A\tan^{-1}B-1}\\ & =\frac{\tan^{-1}A+\tan^{-1}B+\tan^{-1}C-\tan^{-1}A\tan^{-1}B\tan^{-1}C}{1-\tan^{-1}A\tan^{-1}B-\tan^{-1}B\tan^{-1}C-\tan^{-1}C\tan^{-1}A} \end{align*}