Question

Prove that : 4cosxcos(π/3+x)cos(π/3-x) = cos3x

Answer 1

Take L.H.S,

4cosxcos(π/3+x)cos(π/3-x)

= 2 cos x {cos 120° + cos 2x} 

= 2 cos x {cos (180° - 60°) + cos 2x} 

{cos (180° - A) = - cos A} 

= 2 cos x (cos 2x - cos 60°) 

= 2 cos 2x cos x – 2 cos x cos 60°

= { cos(x + 2x) + cos(2x - x)} - \(\frac{2\,cos\,x}{2}\)

= cos 3x + cos x – cos x 

= cos 3x 

= R.H.S. 

Hence Proved.