Question

Y= sin^4x+cos^4x, если tg x =2

Answer 1

Помни, я написал всё, что мог.
Результат может быть ошибочным

Answer 2

Given, tg x = 2.

We can start by using the identity 1 + tan^2(x) = sec^2(x) to find cos(x) and sin(x):

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tan(x) = sin(x) / cos(x)
1 + tan^2(x) = sec^2(x)
1 + (sin^2(x) / cos^2(x)) = 1 / cos^2(x)
cos^2(x) + sin^2(x) = cos^2(x) / cos^2(x)
cos^2(x) + sin^2(x) = 1
sin^2(x) = 1 - cos^2(x)

Substituting tg x = 2 into the equation above:

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tg^2(x) + 1 = sec^2(x)
4 + 1 = sec^2(x)
sec(x) = sqrt(5)

Since sec(x) = 1/cos(x), we have:

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cos(x) = 1/sqrt(5)

Using the identity sin^2(x) = 1 - cos^2(x), we have:

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sin^2(x) = 1 - (1/sqrt(5))^2
sin^2(x) = 1 - 1/5
sin^2(x) = 4/5
sin(x) = sqrt(4/5) = 2/sqrt(5)

Therefore,

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Y = sin^4(x) + cos^4(x)
Y = (2/sqrt(5))^4 + (1/sqrt(5))^4
Y = 16/25 + 1/25
Y = 17/25

Hence, Y = 17/25 when tg x = 2.