Question

if a+b+c=180 degree than prove that cos2a+cos 2b -cos2c= 1-4 sin a sin b cos c

Answer 1

to find sin (x/2), we use the following half angle formula 

sin (x/2) = + or - SQRT [ (1 - cos x) / 2 ] 

Since Pi < x < Pi / 2 then Pi / 2 < x / 2< Pi / 4 so that x/2 is in quadrant 1 and sin (x/2) is positive. Hence 

sin (x/2) = SQRT [ (1 - cos x) / 2 ] 

 Given that sin (x) = 1 / 4, we use the trigonometric identity sin 2x + cos 2x = 1 to find cos x, noting that x is in quadrant 2 and cos x is negative. 

cos x = - sqrt(1 - sin 2x) 

= - sqrt(1 - 1/16) = - sqrt(15) / 4 

We now substitute cos x by its value in the formula for sin (x/2). 

sin (x/2) = SQRT [ (1 - sqrt(15) / 4) / 2 ] 

Which may be simplified to 

= (1/4) SQRT [ 8 - 2 SQRT(15) ]