AP EAMCET202317 May 2023Morning ShiftMathematicsTrigonometric Ratios & IdentitiesActual
In ABC , 2 ~A + 2 ~B + 2 C A + B + C -1 =
Options
- A2[ A+ B+ C]
- BA+ B+ C
- C4[ A+ B+ C]
- D8[ A+ B+ C]
Correct answer
A. 2[ A+ B+ C]
Step-by-step solution
aligned & 2 A+ 2 B+ 2 C A+ B+ C-1 & = 2 ( A +B) ( A - B )+2 Ccos C 2 ( A+B 2 ) ( A-B 2 )+ (1-2 ^2 C 2 )-1 & = 2 C( ( A - B )+ C) 2 C 2 [ ( A-B 2 )- ( A+B 2 ) ] & = C( ( A - B )+ (A+B)) 2 C 2 B 2 A 2 & = 2 C B A 2 C 2 B 2 A 2 & =8 A 2 B 2 C 2 ...(i) aligned Now, A+ B+ C aligned & =2 ( A+B 2 ) ( A-B 2 )+ C & =2 ( C 2 ) [ ( A-B 2 )+ ( 2 - ( A+B 2 ) ) ] & =2 ( C 2 ) [ ( A-B 2 )+ ( A+B 2 ) ] & =4 A 2 B 2 C 2 ...(ii) aligned From equations (i) & (ii), we get 2 A+ 2 B+ 2 C A+ B+ C =2[ A+ B+ C]