Question

(ctg(a/3)-tg(a/3))*tg(2a/3)

Answer 1

Ответ:

соt(a/3) =1/tg(a/3)

теперь поменяем cot на tg

1/tg(a/3)-tg(a/3)×tg(2a/3)=

1-tg(a/3)^2×tg(2a/3) и всё это делить на tg(a/3)

Answer 2

To simplify the given expression $(\cot(\frac{a}{3}) - \tan(\frac{a}{3})) \cdot \tan(\frac{2a}{3})$, we can use trigonometric identities. First, let's simplify the expression inside the parentheses:

$\cot(\frac{a}{3}) - \tan(\frac{a}{3})$

To simplify this, we can use the following identity:

$\cot(x) - \tan(x) = \frac{1 - \tan(x)\cot(x)}{\tan(x) + \cot(x)}$

In this case, $x = \frac{a}{3}$, so we have:

$\frac{1 - \tan(\frac{a}{3})\cot(\frac{a}{3})}{\tan(\frac{a}{3}) + \cot(\frac{a}{3})}$

Next, the expression becomes:

$\frac{1 - \tan(\frac{a}{3})\cot(\frac{a}{3})}{\tan(\frac{a}{3}) + \cot(\frac{a}{3})} \cdot \tan(\frac{2a}{3})$

Now, let's simplify $\tan(\frac{2a}{3})$. We know that $\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$. In this case, $x = \frac{a}{3}$, so:

$\tan(\frac{2a}{3}) = \frac{2\tan(\frac{a}{3})}{1 - \tan^2(\frac{a}{3})}$

Now, substitute this back into the expression:

$\frac{1 - \tan(\frac{a}{3})\cot(\frac{a}{3})}{\tan(\frac{a}{3}) + \cot(\frac{a}{3})} \cdot \frac{2\tan(\frac{a}{3})}{1 - \tan^2(\frac{a}{3})}$

To simplify further, you can multiply the numerators and denominators and simplify the expression. However, it's important to note that the expression might not simplify into a neat and simple form, depending on the specific values of $a$.