Question

if A+B+C=180,prove that sin2A +sin2 +sinC =4SIN A sinB sinC​

Answer 1

Answer:

$answer$

$a + b + c = 180$

$lhs \: = \: sin2a \: + \: sin2b \: + sin2c$

$= 2sin \: (a + b) \: cos \:( a - b) +2sin \: c \: cos \: c$

$= 2sin \: c \: cos \: (a - b) + 2sin \: c \: cos \: c$

$= 2sin \: c( \: cos \: (a - b) + cos \: c)$

$= 2 \: sin \: c( \: cos(a \: - \: b) - \: cos(a + b)$

$= 2sin \: c2sin \: a \: sin \: b$

$= 4 \: sin \: a \: sin \: b \: sin \: c$

$= rhs$

Answer 2

Answer:

answer

a + b + c = 180a+b+c=180

lhs \: = \: sin2a \: + \: sin2b \: + sin2clhs=sin2a+sin2b+sin2c

= 2sin \: (a + b) \: cos \:( a - b) +2sin \: c \: cos \: c=2sin(a+b)cos(a−b)+2sinccosc

= 2sin \: c \: cos \: (a - b) + 2sin \: c \: cos \: c=2sinccos(a−b)+2sinccosc

= 2sin \: c( \: cos \: (a - b) + cos \: c)=2sinc(cos(a−b)+cosc)

= 2 \: sin \: c( \: cos(a \: - \: b) - \: cos(a + b)=2sinc(cos(a−b)−cos(a+b)

= 2sin \: c2sin \: a \: sin \: b=2sinc2sinasinb

= 4 \: sin \: a \: sin \: b \: sin \: c=4sinasinbsinc

= rhs=rhs