Question

cos square A + cos square b + cos square C is equals to 1 - 2 cos a cos b cos c​

Answer 1

Answer:Given: a + b + c = 180° = π

To Prove: cos² a + cos² b - cos² c = 1 - 2 . sin a . sin b . sin c

We use the following formula to prove it,

1 .cos² x - sin² y = cos( x + y ) . cos( x - y )

2. sin² x + cos² y = 1

3. cos ( π - x ) = -cos x

4. cos x - cos y  = 2 . sin (x+y/2) . sin (x-y/2)

5. cos (-x) = cos x

Consider,

cos² a + cos² b - cos² c

= cos² a + 1 - sin² b - cos² c         ( from 2 )

= 1 + cos² a - sin² b - cos² c

= 1 + cos ( a + b ) . cos ( a - b ) - cos² c          ( from 1 )

= 1 + cos ( π - c ) . cos ( a - b ) - cos² c

= 1 - cos c . cos ( a - b ) - cos² c          ( from  3 )

= 1 - cos c ( cos ( a - b ) + cos c )

= 1 - cos c ( cos ( a - b ) + cos ( π - (a+b) ) )

= 1 - cos c ( cos ( a - b ) - cos ( a + b ) )         ( from 3 )

= 1 - cos c ( cos ( a - b ) - cos ( a + b ) )

  ( from 4 )

= 1 - cos c ( 2 . sin a . sin b )                  ( from 5 )

= 1 - 2 . cos a . sin b . sin c

= RHS

Hence Proved

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Answer 2

R.H.S :- SEE THE ABOVE PICTURE OF THE SOLUTION