Grade 12th passTrigonometry

If A+B+C = π then prove that cos 2A + cos 2B - cos 2C =1-4sinAsinBcosC

twisha

8 Years agoGrade 12th pass

1 Answer

Arun

Approved Tutor Answer8 Years ago

cos2A + cos2B - cos2C

= cos2A - 2sin[( 2B + 2C)/2]sin[(2B - 2C)/2]

= cos2A - 2sin( B + C )sin(B - C)

We have : A + B + C = 180° => B + C = 180° - A

= cos2A - 2sin( 180° - A )sin(B - C)

= cos2A - 2( sin180°cosA - sinAcos180° )sin( B - C )

= cos2A + 2sinAsin(B - C)

= 1 - 2sin²A + 2sinAsin(B - C)

= 1 - 2sinA[ sinA - sin(B - C) ]

= 1 - 2sinA( 2cos[(A + B - C)/2]sin[(A - B + C)/2] )

= 1 - 4sinA.cos( A + B - C )/2.sin(A - B + C ) /2

We have : A + B + C = 180° => A + B = 180° - C
And : A + C = 180° - B

= 1 - 4sinA.cos[( 180° - C - C )/2].sin[( 180° - B - B )/2]

= 1 - 4sinA.cos( -2C )/2.sin( -2B)/2

= 1 - 4sinA.cos(-C).sin(-B)

= 1 - 4sinA.sinB.cosC