Prove the following: cot 4x(sin 5x + sin 3x) = cot x(sin 5x - sin 3x)

Solution:

LHS = cot 4x(sin 5x + sin 3x)

= cot 4x [2sin (5x + 3x)/2 cos (5x - 3x)/2] 

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

= (cos 4x / sin 4x) [2 sin 4x cos x]

= 2cos 4x cos x

= 2cos 4x cosx × (sin x / sin x) 

[multiplied and divided by sin x]

= (cos x / sin x) × [2cos 4x sin x]

= cot x [sin (4x + x) - sin (4x - x)]

[Using product to sum formulas, 2 cos A sin B = sin (A + B) - sin (A - B)]

= cot x(sin 5x - sin 3x)

= RHS

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NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.3 Question 15

Prove the following: cot 4x(sin 5x + sin 3x) = cot x(sin 5x - sin 3x)

Summary:

We got, cot 4x(sin 5x + sin 3x) = cot x(sin 5x - sin 3x). Hence Proved