We have to prove that cos 4x - sin 4x * cot 2x = -1

cos 4x - sin 4x * cot 2x = -1

use cos 2x = (cos x)^2 - (sin x)^2 and sin 2x = 2 sin x cos x and cot x = (cos x)/(sin x)

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We have to prove that cos 4x - sin 4x * cot 2x = -1

cos 4x - sin 4x * cot 2x = -1

use cos 2x = (cos x)^2 - (sin x)^2 and sin 2x = 2 sin x cos x and cot x = (cos x)/(sin x)

=> (cos 2x)^2 - (sin x)^2 - 2*(sin 2x)*(cos 2x)*(cos 2x)/(sin 2x)

=>(cos 2x)^2 - (sin x)^2 - 2*(cos 2x)*(cos 2x)

=> ( cos 2x)^2 - 2 ( cos 2x)^2 - ( sin 2x)^2

=> - (cos 2x)^2 - (sin 2x)^2

=> -1*[(cos 2x)^2 + (sin 2x)^2]

As (cos x)^2 + (sin x)^2 = 1

=> -1

Therefore we proved that

**cos 4x - sin 4x * cot 2x = -1**