Question

(1/tgx)-1=cos2x/(1+tgx) - решите уравнение с помощью подстановки y=tgx

Answer 1

Если мы делаем замену y = tg x, то нам надо выразить
$cos 2x = cos^2 x - sin^2 x = cos^2x*(1- \frac{sin^2x}{cos^2x} )=cos^2 x*(1 - tg^2 x) = \\ = \frac{1}{1+tg^2x}*(1-tg^2x) =\frac{1 - tg^2 x}{1 + tg^2 x}$
Подставляем
$\frac{1}{y} -1= \frac{(1-y^2)/(1+y^2)}{1+y}$
$\frac{1-y}{y} = \frac{(1-y)(1+y)}{(1+y^2)(1+y)} = \frac{1-y}{1+y^2}$
$\frac{1-y}{y} - \frac{1-y}{1+y^2}=0$
$\frac{(1-y)(1+y^2)-(1-y)*y}{y(1+y^2)} =0$
Если дробь равна 0, то числитель равен 0, а знаменатель нет
(1 - y)(y^2 - y + 1) = 0
y = tg x = 1; x = pi/4 + pi*k
y^2 - y + 1 = 0 - это уравнение корней не имеет.
Ответ: pi/4 + pi*k

Answer 2

To solve the equation (1/tgx) - 1 = cos(2x) / (1 + tgx) using the substitution y = tgx, follow these steps:

Step 1: Apply the substitution y = tgx Replace tgx in the equation with y:

(1/y) - 1 = cos(2x) / (1 + y)

Step 2: Express cos(2x) in terms of y We know that cos(2x) = 2 * cos^2(x) - 1. Using this trigonometric identity and the fact that tgx = y, we can express cos(2x) in terms of y:

cos(2x) = 2 * (1 - y^2) - 1 cos(2x) = 2 - 2y^2 - 1 cos(2x) = 1 - 2y^2

Step 3: Substitute the expression for cos(2x) into the equation Now, replace cos(2x) in the equation with 1 - 2y^2:

(1/y) - 1 = (1 - 2y^2) / (1 + y)

Step 4: Solve for y To solve for y, we need to get rid of the fractions. Multiply both sides by y(1 + y) to eliminate the denominators:

y(1 + y) * ((1/y) - 1) = y(1 + y) * ((1 - 2y^2) / (1 + y))

Now, simplify the equation:

(1 + y) - y = y(1 - 2y^2)

Expand and rearrange the equation:

1 + y - y - y^2 = y - 2y^3

Combine like terms:

1 - y^2 = 2y - 2y^3

Step 5: Bring all terms to one side and factor Move all terms to one side of the equation:

2y^3 - y^2 - 2y + 1 = 0

Step 6: Factor the equation Now, try to factor the equation. After some trial and error, you can find the factorization:

(y - 1)^2 * (2y + 1) = 0

Step 7: Set each factor to zero and solve for y Set each factor to zero and solve for y:

1. y - 1 = 0 y = 1

2. 2y + 1 = 0 2y = -1 y = -1/2

Step 8: Find x using the original substitution Now that we have the values for y, we can find x using the original substitution y = tgx.

For y = 1: tgx = 1 x = arctan(1) x = π/4 + πk (where k is an integer)

For y = -1/2: tgx = -1/2 x = arctan(-1/2) x = -π/6 + πk (where k is an integer)

So, the solutions to the original equation are: x = π/4 + πk and x = -π/6 + πk, where k is an integer.