Ex 3.3, 15 - Prove cot 4x (sin 5x + sin 3x) = cot x (sin 5x - cos x + cos y formula

  1. Chapter 3 Class 11 Trigonometric Functions
  2. Serial order wise

Transcript

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 (π‘₯ + 𝑦)/2 cos (π‘₯ βˆ’ 𝑦)/2 Putting x = 5x & y = 3x = cot 4x Γ— [ 2 sin ((5π‘₯ + 3π‘₯)/2) cos ((5π‘₯ βˆ’ 3π‘₯)/2) ] = cot 4x Γ— [ 2sin (8π‘₯/2) cos (2π‘₯/2)] = 2 cot 4x sin 4x cos x = 2 π‘π‘œπ‘ β‘4π‘₯/𝑠𝑖𝑛⁑4π‘₯ Γ— sin 4x Γ— cos x = 2 cos 4x cos x Solving R.H.S. cot x ( sin 5x – sin 3x ) Using sin x – sin y = 2 cos (π‘₯ + 𝑦)/2 sin (π‘₯ βˆ’ 𝑦)/2 Putting x = 5x & y = 3x = cot x ( 2 cos (5π‘₯ + 3π‘₯)/2 sin (5π‘₯ βˆ’ 3π‘₯)/2 ) = cot x ( 2 cos (8π‘₯/2) sin (2π‘₯/2)] = cot x ( 2 cos 4x sin x) = 2 cos 4x sin x cot x = 2 cos 4x sin x Γ— π‘π‘œπ‘ β‘π‘₯/sin⁑π‘₯ = 2 cos 4x cos x = L.H.S Hence L.H.S. = R.H.S. Hence Proved

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