Question

Tg x + tg2x /1-tgx tg2x = 1

Answer 1

$\displaystyle \frac{tgx+ tg2x}{1-tgx*tg2x}=1$

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вспомним формулу

$\displaystyle tg(a+b)= \frac{tga+tgb}{1-tga*tgb}$

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тогда левую часть преобразуем по этой формуле

$\displaystyle tg(x+2x)=1\\\\tg3x=1\\\\3x= \frac{ \pi }{4}+ \pi n; n\in Z\\\\x= \frac{ \pi }{12}+ \frac{ \pi n}{3}; n\in Z$

Answer 2

To solve the equation tg(x) + tg(2x) / (1 - tg(x) tg(2x)) = 1, we can simplify it step by step. Let's break it down:

Step 1: Substitute trigonometric identities We'll use the identity for tan(2x) and rearrange the equation: tg(x) + 2tg(x) / (1 - tg(x) * [2tg(x) / (1 - tg^2(x))]) = 1

Step 2: Simplify the expression Next, we'll simplify the expression within the brackets: tg(x) + 2tg(x) / (1 - 2tg^2(x) / (1 - tg^2(x))) = 1

Step 3: Further simplify Now, we'll simplify the expression within the second set of brackets: tg(x) + 2tg(x) / (1 - 2tg^2(x) / (1 - tg^2(x))) = tg(x) + 2tg(x) / (1 - 2tg^2(x) + tg^2(x)) / (1 - tg^2(x)) = tg(x) + 2tg(x) / (1 - tg^2(x) - 2tg^2(x) + tg^4(x)) / (1 - tg^2(x))

Step 4: Combine the fractions To combine the fractions, we'll multiply the first fraction by the reciprocal of the second fraction: [tg(x) + 2tg(x)] * [(1 - tg^2(x)) / (1 - tg^2(x) - 2tg^2(x) + tg^4(x))] = 1

Step 5: Expand and simplify Expanding the numerator, we get: [tg(x) + 2tg(x)] * (1 - tg^2(x)) = 1 - tg^2(x) - 2tg^2(x) + tg^4(x)

Multiplying through, we have: tg(x) - tg^3(x) + 2tg(x) - 2tg^3(x) = 1 - tg^2(x) - 2tg^2(x) + tg^4(x)

Step 6: Combine like terms Collecting like terms, we get: 3tg(x) - 3tg^3(x) = 1 - 3tg^2(x) + tg^4(x)

Step 7: Rearrange the equation Rearranging the equation to set it equal to zero, we have: tg^4(x) - 3tg^3(x) + 3tg^2(x) + 3tg(x) - 1 = 0

This is a quartic equation in terms of tg(x), and finding an exact solution can be quite challenging. However, it can be solved numerically using approximation methods or software.