(cos 8x)/(sin8x-tg4x)

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вопрос

To simplify the expression (cos 8x)/(sin 8x - tan 4x), we can use trigonometric identities and formulas:

cos 8x = 2cos^2(4x) - 1
sin 8x = 2sin(4x)cos(4x)
tan 4x = sin(4x)/cos(4x)

Substitute these identities into the expression:

(cos 8x)/(sin 8x - tan 4x)
= (2cos^2(4x) - 1) / (2sin(4x)cos(4x) - sin(4x)/cos(4x))
= (2cos^2(4x) - 1) / (sin(4x)(2cos(4x) - 1/cos(4x)))
= (2cos^2(4x) - 1) / (sin(4x)(2cos(4x)^2 - 1))
= (2cos^2(4x) - 1) / (sin(4x)(2 - sin(4x)^2))

Now we can simplify further by using trigonometric identities:

cos^2(4x) = (1 + cos(8x))/2
sin^2(4x) = (1 - cos(8x))/2

Plug these identities into the expression:

= [2(1 + cos(8x))/2 - 1] / [sin(4x)(2 - (1 - cos(8x))/2)]
= (1 + cos(8x) - 1) / [2sin(4x) - sin(4x - cos(8x)/2)]
= cos(8x) / [2sin(4x) - sin(4x - cos(8x)/2)]

Therefore, the simplified expression is cos(8x) / [2sin(4x) - sin(4x - cos(8x)/2)].