Express each of the following as an algebraic sum of sines or cosines :

(i) 2sin 6x cos 4x

(ii) 2cos 5x din 3x

(iii) 2cos 7x cos 3x

(iv) 2sin 8x sin 2x

i) 2sin 6x cos 4x = sin (6x+4x) + sin (6x-4x)

= sin 10x + sin 2x

Using,

2sinAcosB = sin (A+ B) + sin (A - B)

ii) 2cos 5x sin3x = sin (5x + 3x) – sin (5x – 3x)

= sin8x – sin2x

Using,

2cosAsinB = sin(A + B) – sin (A - B)

iii) 2cos7xcos3x = cos (7x+3x) + cos (7x – 3x)

=cos10x + cos 4x

Using,

2cosAcosB = cos (A+ B) + cos (A - B)

iv)2sin8xsin2x = cos (8x - 2x) – cos (8x + 2x)

= cos6x – cos10x

Using,

2sinAsinB = cos (A - B) – cos (A+ B)