If-A-B-C-pi-then-sin-2-A-sin-2-B-sin-2-C-2-cos-A-cos-B-cos-C-

Question Number 95515 by bagjamath last updated on 25/May/20

If A + B + C = π, then

\sin^{2} A + \sin^{2} B + \sin^{2} C - 2 \cos A \cos B \cos C =

Answered by som(math1967) last updated on 25/May/20

\sin^{2} A + \sin^{2} B + \sin^{2} C - 2 cosAcos Bcos C

= \frac{1}{2} ({2 \sin}^{2} A + 2 \sin^{2} B + {2 \sin}^{2} C)

- 2 cosAcos Bcos C

= \frac{1}{2} (1 - \cos 2 A) + \frac{1}{2} (1 - \cos 2 B)

1 - \cos^{2} C - 2 cosAcosBcos C

= 2 - \frac{1}{2} 2 \cos (A + B) \cos (A - B)

- \cos^{2} C - 2 cosAcosBcosC

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

- 2 cosAcosBcosC ★

= 2 + cosC {\cos (A - B) - cosC}

- 2 cosAcosBcosC

= 2 + cosC {\cos (A - B) + \cos (A + B)}

- 2 cosAcosBcosC

= 2 + 2 cosAcosBcosC - 2 cosAcosBcosC

= 2 ans

★ A + B + C = π

∴ \cos (A + B) = - cosC

Commented by peter frank last updated on 25/May/20

thank you