Proof:

We have to prove that:

(cos 2x * cos 3x - cos 2x * cos 7x cos x * cos 10 x) /( sin 4x * sin 3x - sin 2x * sin 5x sin 4x * sin 7x) = cot 6x * cot 5x

Let's simplify the left-hand side:

(cos 2x * cos 3x - cos 2x * cos 7x cos x * cos 10 x) /( sin 4x * sin 3x - sin 2x * sin 5x sin 4x * sin 7x)

= [(cos 2x * cos 3x) - (cos 2x * cos 7x) - (cos x * cos 10 x)] / [(sin 4x * sin 3x) - (sin 2x * sin 5x) - (sin 4x * sin 7x)]

= [(cos 2x) * (cos 3x - cos 7x) - (cos x) * (cos 10 x)] / [(sin 4x) * (sin 3x - sin 7x) - (sin 2x) * (sin 5x - sin 7x)]

= [(cos 2x) * (-2 sin 5x sin 2x) - (cos x) * (2 sin 5x sin 2x)] / [(sin 4x) * (-2 sin 2x sin 1x) - (sin 2x) * (-2 sin 5x sin 2x)]

= [(cos 2x) * (-2 sin 5x) - (cos x) * (2 sin 5x)] / [(sin 4x) * (-2 sin 1x) - (sin 2x) * (-2 sin 5x)]

= [(cos 2x - cos x) * (2 sin 5x)] / [(sin 2x) * (2 sin 5x - sin 1x)]

= [(cos 2x - cos x) / (sin 2x)] * [(sin 1x) / (2 sin 5x - sin 1x)]

= [(2 sin x sin x) / (2 sin x cos x)] * [(sin 1x) / (2 sin 5x - sin 1x)]

= [tan x] * [(sin 1x) / (2 sin 5x - sin 1x)]

Now, let's simplify the right-hand side:

cot 6x * cot 5x = (cos 6x / sin 6x) * (cos 5x / sin 5x)

= (cos 6x cos 5x) / (sin 6x sin 5x)

= [(cos 6x cos 5x) / (sin 6x cos 5x)] * [(sin 6x cos 5x) / (sin 6x sin 5x)]

= [cot 6x] * [(sin 30x) / (sin 6x sin 5x)]

= [cot 6x] * [(1/2) /

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