If-A-B-C-pi-Prove-that-cos-2A-cos-2B-cos2C-1-4cosAcos-Bcos-C-

Question Number 191474 by Spillover last updated on 24/Apr/23

If A + B + C = π

Prove that

\cos 2 A + \cos 2 B + \cos 2 C + 1 = - 4 cosAcos Bcos C

Commented by Tinku Tara last updated on 24/Apr/23

You probably need precondition

that A + B + C = π

Answered by Spillover last updated on 25/Apr/23

2 \cos (A + B) \cos (A - B) + 2 \cos^{2} C - 1 + 1

A + B + C = π A + B = π - C

2 \cos (π - C) \cos (A - B) + 2 \cos^{2} C

2 \cos Ccos (A - B) + 2 \cos^{2} C

2 \cos C [\cos (A - B) - \cos C]

C = π - (A + B)

- 2 \cos C [\cos (A + B) + \cos (A - B)]

- 2 \cos C [2 cosAcos B]

- 4 \cos CcosAcos B