Rewriting fractions

[shamieh]

How can I rewrite \(\displaystyle \int \frac{tan^3x}{cos^3x} \, dx\) to \(\displaystyle \int tan^3x sec^3x \, dx\)

What is the identity they are using to do this?

 

[Evgeny.Makarov]

What is the identity they are using to do this?

The definition of $\sec x$.

 

[shamieh]

The definition of $\sec x$.

What??? What do you mean?

\(\displaystyle \sec x \, dx = ln|\sec x + \tan x| + c\)

 

[Evgeny.Makarov]

I am saying they used the definition of $\sec x$ to rewrite the first expression in post #1 to the second one. This is not related to integration; they rewrote purely the function being integrated.

 

[shamieh]

Yes, but \(\displaystyle secx = \frac{1}{cosx}\) not \(\displaystyle tan^3sec^3x\)

can you show me what's going on?

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even if you re wrote it you would still have tan^3x/sec^3x

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Because you don;t have 1/cosx you have tan^3x/cos^3x

 

[Prove It]

even if you re wrote it you would still have tan^3x/sec^3x

Surely not...

$\displaystyle \begin{align*} \frac{\tan^3{(x)}}{\cos^3{(x)}} &= \tan^3{(x)} \left[ \frac{1}{\cos^3{(x)}} \right] \\ &= \tan^3{(x)} \left[ \frac{1}{\cos{(x)}} \right] ^3 \\ &= \tan^3{(x)} \sec^3{(x)} \end{align*}$

 

[shamieh]

Oh I see now.. I didn't know you could rewrite cos^3x to 1/cosx

 

[Prove It]

Oh I see now.. I didn't know you could rewrite cos^3x to 1/cosx

You can't. But you CAN write $\displaystyle \begin{align*} \frac{1}{\cos^3{(x)}} \end{align*}$ as $\displaystyle \begin{align*} \left[ \frac{1}{\cos{(x)}} \right] ^3 \end{align*}$.

 

[Evgeny.Makarov]

Somehow the following post did not show earlier. I must have accidentally closed it before posting.

By definition,
\[
\frac{1}{\cos x}=\sec x.\]
Taking the cube of both sides,
\[
\frac{1}{\cos^3 x}=\left(\frac{1}{\cos x}\right)^3=(\sec x)^3=\sec^3x.
\]
Multiplying both sides by $\tan^3 x$ we get
\[
\frac{\tan^3 x}{\cos^3 x}=\tan^3 (x)\sec^3x.
\]