# ﻿Conditional Identities – I

## 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).