Prove the following: sin²6x - sin²4x = sin 2x sin 10x

Solution:

LHS = sin²6x - sin²4x

= (sin 6x + sin 4x) (sin 6x - sin 4x)

[Using a² - b² formula]

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

 [Since sin A ± sin B = 2sin [(A ± B) / 2] cos [(A ∓ B) / 2] ]

= [2sin 5x cos x] × [2 sin x cos 5x]

= [2sin 5x cos 5x] × [2 sin x cos x]

= [sin (5x + 5x) + sin (5x - 5x)] × [sin (x + x) + sin (x - x)]

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

= [sin 10x + sin 0] × [sin 2x + sin 0]

= sin 10x × sin 2x

[by trigonometric table sin 0 = 0]

= sin 2x sin 10x

= RHS

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NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.3 Question 12

Prove the following: sin²6x - sin²4x = sin 2x sin 10x

Summary:

We got, sin²6x - sin²4x = sin 2x sin 10x. Hence Proved