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Note that sec⁶x can be written as   [sec²x]³ = [(1 + tan²x)]³

And using the binomial theorem, we have.........

[(1 + tan²x)]³  = [tan²x]³ + 3[tan²x]² + 3[tan²x] + 1    ...   therefore  ....

sec⁶x - tan⁶x =

 [tan²x]³ + 3[tan²x]² + 3[tan²x] + 1  - tan⁶x   =

 tan⁶x  + 3[tan²x]² + 3[tan²x] + 1 - tan⁶x   =

3[tan⁴x] + 3[tan²x] + 1   = which is the same thing as the RHS.....

You can convert a fraction to a percentage by:

fraction --> decimal --> percentage

For the fraction --> decimal stage: Example - 3/4 = 0.75 (Divide to get the decimal) 

For the decimal --> percentage stage: 0.75 = 75% (Multiply 0.75 by 100 to get the % / move the decimal place to the right two times [0.75 is the same as 75/100 or 75%]).

You could also try converting 3/4 to something _ / 100 --> multiply both the numerator and denominator by 25 --> (3*25)/(4*25) = 75/100 (100's make things much easier.) From there, you know that 75/100 --> 75% since the denominator is 100. By doing this, you can avoid the intermediate decimal step and directly head from fraction --> percentage since a denominator of 100 is present.

i think something in the world called a calculator is in the world for this problem of yours!

From a trig identity,

cos(Θ) = cos (-Θ)  therefore

cos (wt-x) = cos(-(wt-x)) = cos(x-wt)      so we have ..  (using the angle difference indentity for the cosine)

Acos(x-wt)  = A[cos(x)cos(wt) + sin(x)sin(wt)]  = Asin(x)sin(wt) + Acos(x)cos(wt)    ....and that = the left hand side 

okay then let me try!

$$\frac{6}{4\frac{1}{2}}$$ 

WOW it worked !thanks!but how do i get the same input for other questions!

Copy the code underneath here.  

Open LaTex and paste it in.   Give it a moment to process it and then press ok

\frac{6}{4\frac{1}{2}}

This will be your output

$$\frac{6}{4\frac{1}{2}}$$

Give it a go.  

y -5y+3(y+6)=32

Our goal is to get one term with "y" on one side and a number on the other side:

First, distribute the 3 by multiplying it to both 'y' and '6': y - 5y + 3y + 18 = 32.

Second, combine like terms (the terms with "y") and subtract 18 from both sides to get a number on the right side: 

(1)y - 5y + 3y = -1y = -y

32 - 18 = 14

Therefore, we have -y = 14 --> divide both sides by "-1" to obtain y = -14.

Do a check to see if the equation makes sense.

y -5y+3(y+6)=32; plug in y = -14:

-14 - 5(-14) + 3(-14 + 6) = 32 --> 

-14 + 70 +3(-8) = 32 -->

56 - 24 = 32 -->

32 = 32 --> Yes.

Ummm Melody!Does that mean that i will have to write the whole thing like 

\\\frac{Flour}{Milk}=\frac{1}{1\frac{2}{4}}=\frac{flour\;needed}{2\frac{1}{2}}\\\\\\

this!

is that what u all write to show the fraction one upon others!

Sorry Rosala I misunderstood your question

\\\frac{Flour}{Milk}=\frac{1}{1\frac{2}{4}}=\frac{flour\;needed}{2\frac{1}{2}}\\\\\\

\begin{array}{rll} \frac{1}{1\frac{2}{4}}&=&\frac{flour\;needed}{2\frac{1}{2}}\\\\

2\frac{1}{2}\times \frac{1}{1\frac{2}{4}}&=&2\frac{1}{2}\times \frac{flour\;needed}{2\frac{1}{2}}\\\\

2\frac{1}{2}\times \frac{1}{1\frac{2}{4}}&=&\;flour\;needed\\\\

flour\;needed&=&\;2\frac{1}{2}\div {1\frac{2}{4}}\\\\

\end{array}

This is the LaTex that I used.  It looks horrible when it is in a big chunk like this but you can learn to do it a little a time.  It is not that hard.

---------------------------------------------------------------------

Just look at this bit

\frac{flour\;needed}{2\frac{1}{2}}  

$$\frac{flour\;needed}{2\frac{1}{2}}$$   this is the LaTex output

\ means that a function will follow

frac means fraction  the top is in the first parenthesis { } and the bottom is in the next parenthesis { }

But you can see I have embedded another fraction inside second lot of parenthesis so there will be a fraction on the bottom.

Does that make sense?

no i mean what function u use to get it one upon the other!like when i type a fraction i latex i cant get it actually as a fraction one upon the other!

Well it is a ratio question and they can be done like fractions.

The ratio of flour to milk will always be the same and this ratio is

$$\begin{array}{rll}
flour&:&milk\\
1&:&1\frac{2}{4}\qquad \mbox{this one you have been given}\\
flour\; needed&:&2\frac{1}{2}\qquad \mbox{This is the new ratio but it must be the same as the other one so I turned them into fractions}\\\\
\frac{1}{1\frac{2}{4}}&=&\frac{flour\;needed}{2\frac{1}{2}}
\end{array}$$

Do you understand now Rosala?

Melody what did u use to get that flour/milk  one upon the other??

$$\\\frac{Flour}{Milk}=\frac{1}{1\frac{2}{4}}=\frac{flour\;needed}{2\frac{1}{2}}\\\\\\
\begin{array}{rll}
\frac{1}{1\frac{2}{4}}&=&\frac{flour\;needed}{2\frac{1}{2}}\\\\
2\frac{1}{2}\times \frac{1}{1\frac{2}{4}}&=&2\frac{1}{2}\times \frac{flour\;needed}{2\frac{1}{2}}\\\\
2\frac{1}{2}\times \frac{1}{1\frac{2}{4}}&=&\;flour\;needed\\\\
flour\;needed&=&\;2\frac{1}{2}\div {1\frac{2}{4}}\\\\
\end{array}$$

$${\frac{\left({\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{2}}}{{\mathtt{4}}}}\right)}} = {\frac{{\mathtt{5}}}{{\mathtt{3}}}} = {\mathtt{1.666\: \!666\: \!666\: \!666\: \!666\: \!7}}$$

$$\mbox{So it looks like you need }1\frac{2}{3}\;\mbox{cups of flour}$$

Do you mean

   $$\\\sqrt{\frac{4}{9}}=\frac{2}{3}\\\\
OR\\\\
\frac{\sqrt4}{9}=\frac{2}{9}$$

In that case, it's 1 and 2/3 cups of flour. Assuming a linear proportion. In food prep, it is sometimes not a linear proportion.  This probably is.

$$\begin{array}{rlll}
\frac{2x}{4\pi}+\frac{1-x}{2}&=&0\\\\
\frac{x}{2\pi}+\frac{1-x}{2}&=&0\\\\
2\pi \times \left(\frac{x}{2\pi}+\frac{1-x}{2}\right)&=&2\pi\times 0\\\\
x+\pi(1-x)&=&0\\\\
x+\pi-\pi x &=&0\\\\
x(1-\pi )+\pi &=&0\\\\
x(1-\pi )&=&-\pi \\\\
x&=&\frac{-\pi}{(1-\pi )} \\\\
x&=&\frac{\pi}{(\pi-1 )} \\\\

\end{array}$$

so we would solve this like

$$x/7 + 24 = 26$$ (now out here first we will take 24 to the other side and subtract it from 26)

$$x/7 = 26-24$$  (and now simply subtract them)

$$x/7=2$$  ( and now we will bring 7 to the other side and then multiply it by 2 which will then give us our x)

$$x=2*7$$ (and now we will simply multiply them both to get our x)

$$x=14$$  

$$ANSWER$$ 

I hope u've understood by now but if not then pls be free to ask!

I hope this helps!

What function?

The domain is all x values ( that have a corresponding y value)

The range is all the y values (that have at least one corresponding x values)

Min is the smallest y

Max is the greatest y.

How you find them depends on what information that you have.   

it is supposed to say one and two quarters cups of milk

*** one and two quarts cups of milk ***

Strange unit measure. Convert it to imperial minims then we can solve it.

27/7/14

Modular arithmetic

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i am happy you liked them your highness!

Your Well Wisher

 $$SenoritaRosala$$

Hi Ninja,

I just thought I'd pop ideas in here but probably the best place to get more threads from would be my nightly wraps.  Anyway this is your baby so you can choose.   Thank you.

You may even already have them.  

Growth and decay  

Half Life

WOW They look really yummy!

I mean;

These gifts that you bestow upon me are most wondrous,  

I am Queen of Camelot, this fine city be mine, but I am still humbled by your generosity.

Thank you kind Rosala.

Her Royal Highness,

Queen Guinevere.

Melody though its quiet now a days but your just one " End of Day Wrap" makes my day mind blowing!Reading your wrap is like "as if i am seeing the worlds most precious treasure in front of my eyes"!I have always loved reading your wraps!Thank you very much for them!

Here is something for my honourable Queen!

Hope you like it ! They were especially prepared for my royal highness!