Question

prove that:-
$\frac{ \tan(60 ) + 1 }{ \tan(60 ) - 1} = \frac{1 + \cos(30) }{1 - \cos(30) }$
please do it fast and don't spam.
its urgent ☺️​

Answer 1

When displaying fractions in-line, for example \(\frac{3x}{2}\)

you can set a different display style:

\( \displaystyle \frac{3x}{2} \).

This is also true the other way around

\[ f(x)=\frac{P(x)}{Q(x)} \ \ \textrm{and}

\ \ f(x)=\textstyle\frac{P(x)}{Q(x)} \]

FractionsBinomialsEx3.png

The command \displaystyle will format the fraction as if it were in mathematical display mode. On the other side, \textstyle will change the style of the fraction as if it were part of the text.

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Continued fractions

The usage of fractions is quite flexible, they can be nested to obtain more complex expressions.

The fractions can be nested

\[ \frac{1+\frac{a}{b}}{1+\frac{1}{1+\frac{1}{a}}} \]

Now a wild example

\[

a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}}

\]

FractionsBinomialsEx4.png

The second fraction displayed in the previous example uses the command \cfrac{}{} provided by the package amsmath (see the introduction), this command displays nested fractions without changing the size of the font. Specially useful for continued fractions.

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Binomial coefficients

Binomial coefficients are common elements in mathematical expressions, the command to display them in LATEX is very similar to the one used for fractions.

The binomial coefficient is defined by the next expression:

\[

\binom{n}{k} = \frac{n!}{k!(n-k)!}

\]

And of course this command can be included in the normal

text flow \(\binom{n}{k}\).

FractionsBinomialsEx5.png

As you see, the command \binom{}{} will print the binomial coefficient using the parameters passed inside the braces.