Conditional Identities
Some standard Identities in Triangle:
1. sin2A + sin2B + sin2C = 4 sinA sinB sinC
Proof: L.H.S sin2A + sin2B + sin2C
(since sinC + sinD = 2sin (C + D)/2 cos (C – D)/2),
\(\sin 2A+\sin 2B=2\sin \left( \frac{2A+2B}{2} \right)\cos \left( \frac{2A-2B}{2} \right)\),
\(\sin 2A+\sin 2B=2\sin \left( A+B \right)\cos \left( A-B \right)\),
\(\sin 2A+\sin 2B+\sin 2C=2\sin \left( A+B \right)\cos \left( A-B \right)+\sin 2C\) \(\left( \because \ \sin 2x=2\sin x\cos x \right)\),
\(=2\sin \left( A+B \right)\cos \left( A-B \right)+2\sin C\cos C\) \(\left( \begin{align} & \because A+B+C=\pi \\ & \Rightarrow A+B=\pi -C \\\end{align} \right)\),
\(=2\sin \left( \pi -C \right)\cos \left( A-B \right)+2\sin C\cos C\),
\(=2\sin \left( C \right)\cos \left( A-B \right)+2\sin C\cos C\),
\(=2\sin C\left( \cos \left( A-B \right)+\cos C \right)\),
\(=2\sin C\left( \cos \left( A-B \right)+\cos \left( \pi -\left( A+B \right) \right) \right)\),
\(=2\sin C\left( \cos \left( A-B \right)-\cos \left( A+B \right) \right)\) \(\left( \because \ \cos \left( A+B \right)-\cos \left( A-B \right)=2\sin A\sin B \right)\),
\(=2\sin C\times 2\sin A\sin B\),
\(=4\sin A\sin B\sin C\),
Hence proved
2. cosA + cosB + cosC = 1 + 4sin(A/2) sin(B/2) sin(C/2)
Proof: L.H.S cosA + cosB + cosC – 1
\(=2\cos \left( \frac{A+B}{2} \right)\cos \left( \frac{A-B}{2} \right)+\cos C-1\) \(\left( \because \frac{A+B+C}{2}=\frac{\pi }{2} \right)\),
\(=2\cos \left( \frac{\pi }{2}-\frac{C}{2} \right)\cos \left( \frac{A-B}{2} \right)+\cos C-1\),
\(=2\sin \left( \frac{C}{2} \right)\cos \left( \frac{A-B}{2} \right)+\cos C-1\) \(\left( \because \ {{\sin }^{2}}x=\frac{1-\cos 2x}{2} \right)\),
\(=2\sin \left( \frac{C}{2} \right)\cos \left( \frac{A-B}{2} \right)+1-2{{\sin }^{2}}\left( \frac{C}{2} \right)-1\),
\(=2\sin \left( \frac{C}{2} \right)\cos \left( \frac{A-B}{2} \right)-2{{\sin }^{2}}\left( \frac{C}{2} \right)\),
\(=2\sin \left( \frac{C}{2} \right)\left( \cos \left( \frac{A-B}{2} \right)-\sin \left( \frac{C}{2} \right) \right)\),
\(=2\sin \left( \frac{C}{2} \right)\left( \cos \left( \frac{A-B}{2} \right)-\sin \left( \frac{\pi }{2}-\left( \frac{A+B}{2} \right) \right) \right)\),
\(=2\sin \left( \frac{C}{2} \right)\left( \cos \left( \frac{A-B}{2} \right)-\cos \left( \left( \frac{A+B}{2} \right) \right) \right)\),
\(=2\sin \left( \frac{C}{2} \right)\left( 2\sin \frac{A}{2}\sin \frac{B}{2} \right)\),
\(=4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\),
Hence proved.