Question

If $A+B+C=\pi$ then ${\cos }^{2}A+{\cos }^{2}B-{\cos }^{2}C=1-2\sin A\sin B\cos C$.

• A. True
• B. False

Answer 1

The correct option is A True

Given $A+B+C=\pi$

${\cos }^{2}A+{\cos }^{2}B-{\cos }^{2}C={\cos }^{2}A+{\sin }^{2}C-{\sin }^{2}B={\cos }^{2}A+\sin (C+B)\sin (C-B)=1-{\sin }^{2}A+\sin (\pi -A)\sin (C-B)$

$=1-\sin A(\sin A-\sin (B-C\left)\right)=1-\sin A\left(\sin \right(B+C)-\sin (B-C\left)\right)=1-2\sin A\sin B\sin C$