Prove the following: cos²2x - cos²6x = sin 4x sin 8x

Solution:

LHS = cos²2x - cos²6x

= (cos 2x + cos 6x) (cos 2x - cos 6x)

[Using a² - b² formula]

= [2cos {(2x + 6x) / 2} cos {(2x - 6x) / 2}] × [ -2 sin {(2x + 6x) / 2} sin {(2x - 6x) / 2}] 

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

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

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

 [By trigonometric formulas, cos (-A) = cos A and sin (-A) = -sin A]

= [2cos 4x cos 2x] × [2sin 4x sin 2x]

= [2cos 4x sin 4x] × [2cos 2x sin 2x] 

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

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

= [sin 8x + sin 0] × [sin 4x + sin 0]

= sin 8x × sin 4x 

 [by trigonometric table sin 0 = 0]

= sin 4x sin 8x

= RHS

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NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.3 Question 13

Prove the following: cos²2x - cos²6x = sin 4x sin 8x

Summary:

We got, cos²2x - cos²6x = sin 4x sin 8x. Hence Proved