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Question -

cos3 xsin 3x + sin3 x cos 3x = 3/4 sin 4x



Answer -

We know that,

cos 3θ = 4cos3θ – 3cosθ

So, 4 cos3θ = cos3θ + 3cosθ

cos3 θ = [cos3θ + 3cosθ]/4 …… (i)

Similarly,

sin 3θ = 3sin θ – 4sin3 θ

4 sin3θ = 3sinθ – sin 3θ

sin3θ = [3sinθ – sin 3θ]/4 …….. (ii)

Now,

Let us consider LHS:

cos3 x sin 3x + sin3 xcos 3x

Substituting the values from equation (i) and (ii), weget

cos3 x sin 3x + sin3 xcos 3x = (cos 3x + 3 cos x)/4 sin 3x + (3sin x – sin 3x)/4 cos 3x

= 1/4 (sin 3x cos 3x + 3 sin 3x cox x + 3sin x cos 3x– sin 3x cos 3x)

= 1/4 (3(sin 3x cos x + sin x cos 3x) + 0)

= 1/4 (3 sin (3x + x))

(We know, sin(x + y) = sin x cos y + cos x sin y)

= 3/4 sin 4x

= RHS

Hence proved.

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