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Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x - Mathematics

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Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x

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उत्तर

L.H.S. = sin x + sin 3x + sin 5x + sin 7x = (sinx + sin7x) + (sin3x + sin5x)

= `2sin  (7x + x)/2 cos  (7x - x)/2 + 2sin  (5x +3x)/2 cos  (5x - 3x)/2` ∵ `[sinx + sin y = 2sin  (x + y)/2 cos  (x - y)/2]`

= 2sin 4x cos 3x + 2sin 4x cos x  [∵ cos (-θ) = cos θ]

= 2sin 4x (cos 3x + cosx) ∵ `[cos x + cos y = 2cos  (x + y)/2 cos  (x - y)/2]`

= `2sin 4x (2cos  (3x +x)/2  cos  (3x - x)/2)`

= 2 sin 4x (2cos 2x cos x) = 4 cos x cos 2x sin 4x

= R.H.S.

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पाठ 3: Trigonometric Functions - Miscellaneous Exercise [पृष्ठ ८२]

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एनसीईआरटी Mathematics [English] Class 11
पाठ 3 Trigonometric Functions
Miscellaneous Exercise | Q 5 | पृष्ठ ८२

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