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Question Number 21268 by oyshi last updated on 18/Sep/17

$$ifA+B+C=\pi$$$$soproof$$$$\sin A+\sin B+\sin C=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$$

Answered by myintkhaing last updated on 18/Sep/17

$$\mathrm{L}.\mathrm{H}.\mathrm{S}=\sin \mathrm{A}+\sin \mathrm{B}+\sin \mathrm{C}$$$$=2\sin \frac{\mathrm{A}+\mathrm{B}}{2}\cos \frac{\mathrm{A}-\mathrm{B}}{2}+2\sin \frac{\mathrm{C}}{2}\cos \frac{\mathrm{C}}{2}$$$$=2\sin (\frac{\pi }{2}-\frac{\mathrm{C}}{2})\cos \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)+2\sin (\frac{\pi }{2}-\frac{\mathrm{A}+\mathrm{B}}{2})\cos \frac{\mathrm{C}}{2}$$$$=2\cos \frac{\mathrm{C}}{2}\cos \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)+2\cos \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right)\cos \frac{\mathrm{C}}{2}$$$$=2(\cos \frac{\mathrm{A}+\mathrm{B}}{2}+\cos \frac{\mathrm{A}-\mathrm{B}}{2})\cos \frac{\mathrm{C}}{2}$$$$=2\left(2\cos \frac{\mathrm{A}}{2}\cos \frac{\mathrm{B}}{2}\right)\cos \frac{\mathrm{C}}{2}$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$\mathrm{use}$$$$\mathrm{A}+\mathrm{B}=\pi -\mathrm{C}\Rightarrow \frac{\mathrm{A}+\mathrm{B}}{2}=\frac{\pi }{2}-\frac{\mathrm{C}}{2}$$$$\sin \mathrm{x}+\sin \mathrm{y}=2\sin \frac{\mathrm{x}+\mathrm{y}}{2}\cos \frac{\mathrm{x}-\mathrm{y}}{2}$$$$\cos \mathrm{x}+\cos \mathrm{y}=2\cos \frac{\mathrm{x}+\mathrm{y}}{2}\cos \frac{\mathrm{x}-\mathrm{y}}{2}$$

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