Question

a(cos B cosC + cos A) = b (cosC cos A+ cos B ) = c ( cos A cosB + cos C)​

Answer 1

Step-by-step explanation:

How can I prove that cos a+cos b+ cos c+cos (a+b+c) =4Cos ((a+b) /2) cos ((b+c) /2) cos ((c+a) /2)?

cos a + cos b + cos c + cos (a+b+c)

=2 cos ((a+b)/2) cos ((a-b)/2) + 2 cos ((c+a+b+c)/2) cos ((c-a-b+c)/2)

= 2cos((a+b)/2)cos((a-b)/2) + 2cos((a+b+2c)/2)cos((-a-b)/2)

= 2cos((a+b)/2)cos((a-b)/2) + 2cos((a+b+2c)/2)cos((-(a+b))/2)

Remember cos(-x) = cos x and cos a + cos b is 2cos((a+b)/2)

= 2cos((a+b)/2)cos((a-b)/2) + 2cos((a+b+2c)/2)cos((a+b)/2)

2cos((a+b)/2) is taken common

= 2 cos ((a+b)/2) { cos ((a-b)/2) + cos ((a+b+2c)/2) }

= 2 cos ((a+b)/2) { 2cos((a-b+a+b+2c)/2*2) cos(a-b-a-b-2c)/2*2) }

= 2 cos ((a+b)/2) { 2 cos ((2a+2c)/4) cos((-2b-2c)/4) }

= 4 cos ((a+b)/2) cos ((a+c)/2) cos ((b+c)/2)