Question

1 - ctg Альфа -------------------- = - ctg Альфа 1 - tg Альфа Докажите тождество!

Answer 1

Ответ на фото с решением

Answer 2

To prove the given trigonometric identity, we need to manipulate the expressions on both sides of the equation until they are equal.

Given identity: 1 - cot(α) / (1 - tan(α)) = -cot(α)

Step 1: Express everything in terms of sin and cos. cot(α) = 1 / tan(α) = cos(α) / sin(α)

Step 2: Substitute cot(α) in the given identity. 1 - (cos(α) / sin(α)) / (1 - tan(α)) = -(cos(α) / sin(α))

Step 3: Combine fractions in the numerator. 1 - (cos(α) / sin(α)) / (1 - (sin(α) / cos(α))) = -(cos(α) / sin(α))

Step 4: Invert the denominator in the middle term and simplify. 1 - (cos(α) / sin(α)) / ((cos(α) - sin(α)) / cos(α)) = -(cos(α) / sin(α))

Step 5: Multiply the numerator and denominator of the middle term by cos(α). 1 - (cos(α) / sin(α)) * (cos(α) / (cos(α) - sin(α))) = -(cos(α) / sin(α))

Step 6: Simplify the expression in the middle term. 1 - (cos^2(α) / (sin(α) * (cos(α) - sin(α)))) = -(cos(α) / sin(α))

Step 7: Factor out -1 from the right-hand side. 1 - (cos^2(α) / (sin(α) * (cos(α) - sin(α)))) = -1 * (cos(α) / sin(α))

Step 8: Use the Pythagorean identity: sin^2(α) + cos^2(α) = 1. 1 - ((1 - sin^2(α)) / (sin(α) * (cos(α) - sin(α)))) = -1 * (cos(α) / sin(α))

Step 9: Rewrite the expression using common denominators. 1 - (1 - sin^2(α)) / (sin(α) * (cos(α) - sin(α))) = -1 * (cos(α) / sin(α))

Step 10: Combine terms in the numerator of the first term. (sin^2(α) / (sin(α) * (cos(α) - sin(α)))) = -1 * (cos(α) / sin(α))

Step 11: Cancel out sin(α) in the numerator with sin(α) in the denominator. (sin(α) / (cos(α) - sin(α))) = -1 * (cos(α) / sin(α))

Step 12: Multiply both sides by sin(α) to eliminate the denominator. sin(α) = -cos(α) * (cos(α) - sin(α))

Step 13: Distribute the -cos(α) on the right-hand side. sin(α) = -cos^2(α) + cos(α) * sin(α)

Step 14: Move all terms to one side of the equation. sin(α) + cos^2(α) - cos(α) * sin(α) = 0

Step 15: Factor out common terms. sin(α) + cos(α) * (cos(α) - sin(α)) = 0

Step 16: Use the Pythagorean identity: sin^2(α) + cos^2(α) = 1. 1 + cos(α) * (cos(α) - sin(α)) = 0

Step 17: Expand the expression on the left-hand side. 1 + cos^2(α) - cos(α) * sin(α) = 0

Step 18: Rearrange the terms to complete the proof. cos^2(α) - cos(α) * sin(α) + 1 = 0

Now, we have shown that the left-hand side equals zero, and thus we have proven the trigonometric identity:

cos^2(α) - cos(α) * sin(α) + 1 = 0

Or, in another form:

1 - cot(α) / (1 - tan(α)) = -cot(α)