Ctg 5pi-x/2 + 2ctg(pi - x/2)=1

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вопрос

To simplify this expression, we will first rewrite ctg in terms of sine and cosine:

ctg(x) = 1/tan(x) = cos(x)/sin(x)

Now, let's rewrite the given expression using this formula:

5(cos(x)/sin(x)) - 2(cos(pi - x)/sin(pi - x))

Since sin(pi - x) = sin(pi)cos(x) - cos(pi)sin(x) = 0 - (-1)*sin(x) = sin(x),

we can simplify the expression further:

5(cos(x)/sin(x)) - 2(-1)(cos(x)/sin(x))
= 5(cos(x)/sin(x)) + 2(cos(x)/sin(x))
= (5+2)(cos(x)/sin(x))
= 7*(cos(x)/sin(x))

Next, we know that cotangent (ctg) is the reciprocal of tangent (tan), which is equal to 1/(tan(x)) = cos(x)/sin(x). Thus, we have:

7*(cos(x)/sin(x)) = 1

Therefore, the given expression simplifies to 1.