Conditional Identities – I
1. sin2A + sin2B + sin2C = 4 sinA sinB sinC
Proof: sin2A + sin2B + sin2C = 4 sinA sinB sinC
L.H.S
(sin2A + sin2B) + sin2C
= 2sin (A + B) cos (A – B) + sin 2C
A + B + C = π
A + B = π – C
Sin (A + B) = Sin (π – C)
Sin (A + B) = Sin C
2sinC sin C = sin2C
= 2sin (C) cos (A – B) + 2sinC cos C
= 2sin C (cos (A – B) + cos C)
= 2sinC (cos (A – B) – cos (A + B))
= 2 sinC [2 sinA sinB]
= 4 sinA sinB sinC
Hence proved
sin2A + sin2B + sin2C = 4 sinA sinB sinC
2. cos2A + cos2B + cos2C = 1 – 4 cosA cosB cosC
Proof: cos2A + cos2B + cos2C = 1 – 4 cosA cosB cosC
L.H.S
(cos2A + cos2B) + cos2C
= A + B + C = π
A + B = π – C
= 2 cos (A + B) cos (A – B) + 2cos²C – 1
= 2 cos (π – C) cos (A – B) + 2cos²C – 1
= – 2 cosC cos (A -B) + 2cos²C – 1
= -2 cosC [cos (A – B) – cosC] – 1
= -2 cosC [cos (A – B) – cosC] – 1
= A + B + C = π
C = π – (A + B)
= – 2cosC [cos (A – B) – cos (π – (A + B))] – 1
= – 2cosC [cos (A – B) + cos (A + B)] – 1
= – 4 cosC cosA cosB – 1
Hence proved
cos2A + cos2B + cos2C = 1 – 4 cosA cosB cosC
3. cosA + cosB + cosC = 1 + 4 sin (A/2). sin (B/2). sin (C/2)
Proof: cosA + cosB + cosC = 1 + 4 sin(A/2). sin(B/2). sin(C/2)
cosA + cosB + cosC – 1
\(=2\cos \left( \frac{A+B}{2} \right).\cos \left( \frac{A-B}{2} \right)+\cos C-1\),
\(\frac{A+B+C}{2}=\frac{\pi }{2}\),
\(\frac{A+B}{2}=\frac{\pi }{2}-\frac{C}{2}\),
\(=2\cos \left( \frac{\pi }{2}-\frac{C}{2} \right).\cos \left( \frac{A-B}{2} \right)+\cos C-1\),
\(=2\sin \left( \frac{C}{2} \right).\cos \left( \frac{A-B}{2} \right)+\cos C-1\),
\(=2\sin \left( \frac{C}{2} \right).\cos \left( \frac{A-B}{2} \right)+1-2{{\sin }^{2}}\frac{C}{2}-1\),
\(=2\sin \left( \frac{C}{2} \right).\cos \left( \frac{A-B}{2} \right)-2{{\sin }^{2}}\frac{C}{2}\),
\(=2\sin \left( \frac{C}{2} \right)\left( \cos \left( \frac{A-B}{2} \right)-\sin \left( \frac{C}{2} \right) \right)\),
\(=2\sin \left( \frac{C}{2} \right)\left( \cos \left( \frac{A-B}{2} \right)-\sin \left( \frac{\pi }{2}-\left( \frac{A+B}{2} \right) \right) \right)\),
\(=2\sin \left( \frac{C}{2} \right)\left( \cos \left( \frac{A-B}{2} \right)-\cos \left( \left( \frac{A+B}{2} \right) \right) \right)\),
\(=2\sin \left( \frac{C}{2} \right).2\sin \left( \frac{A}{2} \right).\sin \left( \frac{B}{2} \right)\),
\(=4\sin \left( \frac{A}{2} \right).\sin \left( \frac{B}{2} \right).\sin \left( \frac{C}{2} \right)\).
Hence proved
cosA + cosB + cosC = 1 + 4 sin (A/2). sin (B/2). sin (C/2).