Ex 3.3, 12 - Chapter 3 Class 11 Trigonometric Functions

Transcript

Ex 3.3, 12 Prove that sin2 6𝑥 – sin2 4𝑥 = sin⁡2𝑥 sin⁡10𝑥 Solving L.H.S. sin2 6x – sin2 4x = (sin 6x + sin 4x) (sin 6x – sin 4x) Lets calculate (sin 6x + sin 4x) and (sin 6x – sin 4x) separately sin 6x + sin 4x = 2 sin ((6x+4x)/2) cos ((6x−4x)/2) = 2sin (10𝑥/2) cos (2𝑥/2) = 2sin 5x cos x sin 6x – sin 4x = 2 cos ((6x+4x)/2) sin((6x−4x)/2) = 2 cos (10𝑥/2) sin (2𝑥/2) = 2 cos 5x sin x Hence sin2 6x – sin2 4x = (sin 6x + sin 4x) (sin 6x – sin 4x) = (2 sin 5x cos x) (2 cos 5x sin x) = (2 sin 5x cos 5x) . (2 sin x cos x) = (sin 10x) × (sin 2x) = R.H.S. Hence, L.H.S. = R.H.S. Hence proved