Question

10.sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x

Answer 1

Write the expression of L.H.S.

sin( n+1 )x⋅sin( n+2 )x+cos( n+1 )x⋅cos( n+2 )x

Simplify the above expression using identity −2sinA⋅sinB=cos( A+B )+cos( A−B ) and 2cosA⋅cosB=cos( A+B )+cos( A−B )

The expression will be

= 1 2 [ 2sin( n+1 )x⋅sin( n+2 )x+2cos( n+2 )x ] = 1 2 [ cos{ ( n+1 )x−( n+2 )x }−cos{ ( n+1 )x+( n+2 )x }+cos{ ( n+1 )x+( n+2 )x } +cos{ ( n+1 )x−( n+2 )x } ] = 1 2 ×2cos{ ( n+1 )x−( n+2 )x } =cos( x )

Write the expression of R.H.S.

=cos( x )

Hence, sin( n+1 )x⋅sin( n+2 )x+cos( n+1 )x⋅cos( n+2 )x=cos( x ) is proved.