Prove the following: sin 2x + 2sin 4x + sin 6x = 4cos²x sin 4x

Solution:

LHS = sin 2x + 2sin 4x + sin 6x

= [sin 2x + sin 6x] + 2sin 4x

= [2sin (2x + 4x)/2 cos (2x - 6x)/2] + 2sin 4x

 [Since sin A + sin B = 2sin [(A + B) / 2] cos [(A - B) / 2]

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

= 2sin 4x cos 2x + 2 sin 4x 

[because cos (-A) = cos A]

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

= 2sin 4x (2cos²x - 1 + 1) 

 [Using double angle formulas, cos 2x = 2cos²x - 1]

= 2sin 4x (2cos²x)

= 4cos²x sin 4x

= RHS

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NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.3 Question 14

Prove the following: sin 2x + 2sin 4x + sin 6x = 4cos²x sin 4x

Summary:

We got, sin 2x + 2sin 4x + sin 6x = 4cos²x sin 4x, Hence Proved