Question

(1-cosA)(1-cosB)(1-cosC)=(1+cosA)(1+cosB)(1+cosC)then show that each side is equal to +-(sinAsinBsinC)

Answer 1

Step-by-step explanation:

( 1-cosA ) ( 1-cosB ) ( 1-cosC ) = ( 1+cosA ) ( 1+cosB ) ( 1+cosC )

Now multiply the both sides by ( 1-cosA ) ( 1-cosB ) ( 1-cosC )

⇒( 1+cosA ) ( 1-cosA ) ( 1+cosB ) ( 1-cosB ) ( 1+cosC ) ( 1-cosC )= ( 1-cosA )^2 ( 1-cosB )^2 ( 1-cosC )^2

⇒ sin²A sin²B sin²C =  ( 1-cosA )² * ( 1-cosB )² * ( 1-cosC )²

⇒ √ (sin²A sin²B sin²C ) = √ [ ( 1-cosA )² * ( 1-cosB )² * ( 1-cosC )² ] [apply root in both side]

⇒ (sinA * sinB * sinC ) = ( 1-cosA ) ( 1-cosB ) ( 1-cosC )

hence proved each side is equal to +- (sinA sinB sinC)