Question and Answers Forum

All Questions      Topic List

UNKNOWN Questions

Previous in All Question      Next in All Question      

Previous in UNKNOWN      Next in UNKNOWN      

Question Number 95515 by bagjamath last updated on 25/May/20

$$\mathrm{If}A+B+C=\pi ,\mathrm{then}$$$${\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C-2\cos A\cos B\cos C=$$

Answered by som(math1967) last updated on 25/May/20

$${\sin }^2\mathrm{A}+\sin {}^2\mathrm{B}+{\sin }^2\mathrm{C}-2\mathrm{cosAcos}\mathrm{Bcos}\mathrm{C}$$$$=\frac{1}{2}({2\sin }^2\mathrm{A}+2\sin {}^2\mathrm{B}+{2\sin }^2\mathrm{C})$$$$-2\mathrm{cosAcos}\mathrm{Bcos}\mathrm{C}$$$$=\frac{1}{2}(1-\cos 2\mathrm{A})+\frac{1}{2}(1-\cos 2\mathrm{B})$$$$1-\cos {}^2\mathrm{C}-2\mathrm{cosAcosBcos}\mathrm{C}$$$$=2-\frac{1}{2}2\cos (\mathrm{A}+\mathrm{B})\cos (\mathrm{A}-\mathrm{B})$$$$-\cos {}^2\mathrm{C}-2\mathrm{cosAcosBcosC}$$$$=2+\mathrm{cosCcos}(\mathrm{A}-\mathrm{B})-{\cos }^2\mathrm{C}$$$$-2\mathrm{cosAcosBcosC}★$$$$=2+\mathrm{cosC}\{\cos (\mathrm{A}-\mathrm{B})-\mathrm{cosC}\}$$$$-2\mathrm{cosAcosBcosC}$$$$=2+\mathrm{cosC}\{\cos (\mathrm{A}-\mathrm{B})+\cos (\mathrm{A}+\mathrm{B})\}$$$$-2\mathrm{cosAcosBcosC}$$$$=2+2\mathrm{cosAcosBcosC}-2\mathrm{cosAcosBcosC}$$$$=2\mathrm{ans}$$$$★\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$$$$\therefore \cos (\mathrm{A}+\mathrm{B})=-\mathrm{cosC}$$

Commented by peter frank last updated on 25/May/20

$$\mathrm{thank}\mathrm{you}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com