Question

If A + b + C = $\pi$ , then Prove That: $co{s}^{2}A+co{s}^{2}B-co{s}^{2}C=1-2sinA.SinB.cosC$

Answer 1

$\begin{array}{c}A+B+C=\pi \\ provethat:\\ {\cos }^{2}A+{\cos }^{2}B-{\cos }^{2}C=1-\sin A.\sin B.\cos C\\ {\cos }^{2}A+{\sin }^{2}C-{\sin }^{2}B\\ 1-\sin A+\sin A.\cos \left(C-B\right)\\ 1-\sin A\left[-\sin \left(C-B\right)+\sin A\right]\\ 1-\sin A\left[-\sin \left(B+C\right)+\sin \left(B-C\right)\right]\\ 1-\sin A2\sin B\cos C\\ =1-2\sin A\sin B\cos C\\ Hence,\\ proved\end{array}$