Ex 3.3, 15 - Chapter 3 Class 11 Trigonometric Functions

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Ex 3.3, 15 Prove that cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x) Solving L.H.S. cot 4x ( sin 5x + sin 3x ) Using sin x + sin y = 2 sin (𝑥 + 𝑦)/2 cos (𝑥 − 𝑦)/2 Putting x = 5x & y = 3x = cot 4x × [ 2 sin ((5𝑥 + 3𝑥)/2) cos ((5𝑥 − 3𝑥)/2) ] = cot 4x × [ 2sin (8𝑥/2) cos (2𝑥/2)] = 2 cot 4x sin 4x cos x = 2 𝑐𝑜𝑠⁡4𝑥/𝑠𝑖𝑛⁡4𝑥 × sin 4x × cos x = 2 cos 4x cos x Solving R.H.S. cot x ( sin 5x – sin 3x ) Using sin x – sin y = 2 cos (𝑥 + 𝑦)/2 sin (𝑥 − 𝑦)/2 Putting x = 5x & y = 3x = cot x ( 2 cos (5𝑥 + 3𝑥)/2 sin (5𝑥 − 3𝑥)/2 ) = cot x ( 2 cos (8𝑥/2) sin (2𝑥/2)] = cot x ( 2 cos 4x sin x) = 2 cos 4x sin x cot x = 2 cos 4x sin x × 𝑐𝑜𝑠⁡𝑥/sin⁡𝑥 = 2 cos 4x cos x = L.H.S Hence L.H.S. = R.H.S. Hence Proved