(1-tgx)/(1-ctgx)=2sinx

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вопрос

To prove the identity:

(1-tan(x))/(1-cot(x)) = 2*sin(x)

we can start by expressing tan(x) and cot(x) in terms of sine and cosine:

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

cot(x) = cos(x)/sin(x)

Substitute these expressions into the original equation:

(1 - sin(x)/cos(x))/(1 - cos(x)/sin(x))

To simplify this expression, we can first find a common denominator:

(1 - sin(x)/cos(x))/(1 - cos(x)/sin(x)) = ((cos(x) - sin(x))/(cos(x)) / ((sin(x) - cos(x))/(sin(x)))

Next, simplify the expression further:

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

simplify:

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

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

= tan(x) = tan(x)

Therefore, we have verified that the given identity holds true:

(1-tan(x))/(1-cot(x)) = 2*sin(x)