cos 4a - sin 4a + ctg 2a = 1

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вопрос

To simplify the expression, we can use trigonometric identities to rewrite the terms.

First, we know that cos(2a) = cos^2(a) - sin^2(a) and sin(2a) = 2sin(a)cos(a).

Now let's rewrite the expression cos(4a) - sin(4a) + cot(2a):

= (cos(2a))^2 - (sin(2a))^2 + cot(2a)
= (cos^2(a) - sin^2(a))^2 - (2sin(a)cos(a))^2 + cot^2(2a)
= cos^4(a) - 2cos^2(a)sin^2(a) + sin^4(a) - 4sin^2(a)cos^2(a) + cot^2(2a)

Using the trigonometric identity cot(2a) = cos(2a)/sin(2a) = (cos^2(a) - sin^2(a))/(2sin(a)cos(a)), we can simplify it further:

= cos^4(a) - 2cos^2(a)sin^2(a) + sin^4(a) - 4sin^2(a)cos^2(a) + (cos^4(a) - sin^4(a))/(4sin^2(a)cos^2(a))
= 2cos^4(a) - 2sin^4(a) - 4cos^2(a)sin^2(a) + (cos^4(a) - sin^4(a))/(4sin^2(a)cos^2(a))
= 2cos^4(a) - 2sin^4(a) - 4cos^2(a)sin^2(a) + (cos^2(a) - sin^2(a))(cos^2(a) + sin^2(a))/(4sin^2(a)cos^2(a))

Since cos^2(a) + sin^2(a) = 1, we have:

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

The expression cos(4a) - sin(4a) + cot(2a) does not simplify to 1.