Misc 7 - Chapter 3 Class 11 Trigonometric Functions

Transcript

Misc 7 Prove that: sin 3x + sin2x – sin x = 4 sin x cos 𝑥/2 cos 3𝑥/2 Solving L.H.S sin 3x + sin 2x - sin x = sin 3x + (sin 2x – sin x ) = sin 3x + [2cos ((2𝑥 + 𝑥)/2) . sin ((2𝑥−𝑥)/2)] = sin 3x + [ 2cos (3𝑥/2) sin 𝑥/2] = 2sin 3𝑥/2 cos 3𝑥/2 + [ 2 cos 3𝑥/2 sin 𝑥/2 ] = 2cos 3𝑥/2 [ sin 3𝑥/2 + sin 𝑥/2 ] = 2 cos 3𝑥/2 [2 sin ("(" 3𝑥/2 " + " 𝑥/2 ")" )/2 . cos ("(" 3𝑥/2 " − " 𝑥/2 ")" )/2] = 2 cos 3𝑥/2 [ 2 sin ("(" (3𝑥 + 𝑥)/2 ")" )/2 . cos ("(" (3𝑥 − 𝑥)/2 ")" )/2] = 2 cos 3𝑥/2 [ 2 sin ("(" 4𝑥/2 ")" )/2 . cos ("(" 2𝑥/2 ")" )/2] = 2 cos 3𝑥/2 [ 2 sin ("(" 2𝑥/1 ")" )/2 . cos ("(" 𝑥/1 ")" )/2] = 2 cos 3𝑥/2 [ 2 sin 2𝑥/2 . cos 𝑥/2] = 2 cos 3𝑥/2 [ 2 sin 𝑥 . cos 𝑥/2] = 4 cos 3𝑥/2 sin 𝑥 cos 𝑥/2 = R.H.S Hence L.H.S = R.H.S Hence proved