Expression eqivalent to 1+sec/cos ? Explain

[aisha]

Expression eqivalent to 1+sec/cos ?? Explain please

How do u find an expression equivalent to \frac {1+\sec\theta} {\cos\theta} to be \frac {\cos\theta+1} {\cos^2\theta}

When I did this question I got my answer to be \frac {2} {\cos\theta}

Im not very good with identities can someone please explain to me how to do this question and other like it.

 

[learningphysics]

Can you show your steps to getting that? I'll point out where you went wrong.

 

[aisha]

\frac {sin^2\theta + \cos^2\theta} {\cos \theta} + \frac {\sin^2\theta+\cos^2\theta} {\cos\theta} (this is over \cos \theta)

 

[learningphysics]

\frac {sin^2\theta + \cos^2\theta} {\cos \theta} + \frac {\sin^2\theta+\cos^2\theta} {\cos\theta} (this is over \cos \theta)

Forget about sin^2\theta + \cos^2\theta =1.

Do the problem again, but this time only use:sec\theta=\frac{1}{cos\theta} and try to simplify it.

 

[aisha]

I have gone through so many of these types of questions but I have no idea of how to simplify it, I've looked everywhere for a similar example but just can't find anything, I don't know what to do... How do u solve these questions that ask for equivalent identities?

 

[cepheid]

Well, did you take learningphysics' advice? Substitute 1/sec(x) every time you see cos(x) to get the expression in terms of secant, instead of cosine. That is definitely a step in the right direction.

BTW, check back to your thread "please explain". I posted the solution to your previous problem there, since you were very close to understanding it anyway.

 

[dextercioby]

Use:
\sec x =\frac{1}{\cos x}

and
\frac{1+\frac{1}{a}}{b}=\frac{a+1}{ab}

which is an elementary property of fractions.

Daniel.

 

[aisha]

Use:
\sec x =\frac{1}{\cos x}

and
\frac{1+\frac{1}{a}}{b}=\frac{a+1}{ab}

which is an elementary property of fractions.

Daniel.

Can u explain how u got this elementary property of fractions, this might be the step I don't remember lol

 

[learningphysics]

I have gone through so many of these types of questions but I have no idea of how to simplify it, I've looked everywhere for a similar example but just can't find anything, I don't know what to do... How do u solve these questions that ask for equivalent identities?

I plugged in sec\theta=\frac{1}{cos\theta}

I get:

\frac {1+\frac{1}{cos\theta}} {\cos\theta}

Do something to the numerator and denominator so that you get a cleaner looking expression. Try different things... you can try adding the two terms in the numerator together (get a common denominator etc...)

 

[aisha]

I plugged in sec\theta=\frac{1}{cos\theta}

I get:

\frac {1+\frac{1}{cos\theta}} {\cos\theta}

Do something to the numerator and denominator so that you get a cleaner looking expression. Try different things... you can try adding the two terms in the numerator together (get a common denominator etc...)

This is exactly where I am stuck!I tried multiplying the numerator and denominator by cos theta, but that didnt work, I took ur advice and I am half way I made the numerators have a common denominator

so I got 1+\frac{1}{\cos\theta} = \frac {\cos\theta} {\cos\theta}+ \frac {1}{\cos\theta} This is only the numerator of the question my answer was \frac {\cos\theta+1} {\cos\theta} But this is all over \cos\theta what do u do next? How do you get cos^2theta in the denominator of the answer ?

 

[learningphysics]

Can u explain how u got this elementary property of fractions, this might be the step I don't remember lol

Try working on the expression dextercioby gave... how can you rewrite the expression... try experimenting...

That 1/a is messy, how can you rewrite the expression so that the 1/a is no longer there?

 

[dextercioby]

Can u explain how u got this elementary property of fractions, this might be the step I don't remember lol

\frac{1+\frac{1}{a}}{b}=\frac{\frac{a+1}{a}}{b}=\frac{a+1}{ab}

Daniel.

 

[learningphysics]

This is exactly where I am stuck!I tried multiplying the numerator and denominator by cos theta, but that didnt work, I took ur advice and I am half way I made the numerators have a common denominator

so I got 1+\frac{1}{\cos\theta} = \frac {\cos\theta} {\cos\theta}+ \frac {1}{\cos\theta} This is only the numerator of the question my answer was \frac {\cos\theta+1} {\cos\theta} But this is all over \cos\theta what do u do next? How do you get cos^2theta in the denominator of the answer ?

Ok. I see where you are stuck now. The whole fraction is:
\frac { \frac {\cos\theta} {\cos\theta}+ \frac {1}{\cos\theta}}{\cos\theta}

I'll rewrite the numerator, and the fraction becomes:
\frac { \frac {\cos\theta +1}{\cos\theta}}{cos\theta}

Now the denominator of this fraction is \cos\theta

Instead of dividing by \cos\theta, I'm going to multiply by:
\frac{1}{\cos\theta}

So we have:
\frac {\cos\theta +1}{\cos\theta} * \frac{1}{\cos\theta}

Multiply the numerators, and multiply the denominators and we get:
\frac {\cos\theta+1}{\cos^2\theta}

 

[learningphysics]

\frac{(\frac{a}{b})}{c} = \frac{a}{bc}

 

[K.J.Healey]

Why not just multiple top and bottom by cos...
\frac {1+\sec\theta} {\cos\theta}
\frac {1+\sec\theta} {\cos\theta}(\frac{\cos\theta}{\cos\theta})
\frac {\cos\theta+1} {\cos^2\theta}

because cos*sec =1 = cos*(1/cos) = 1

 

[aisha]

WoW I think I get it thanks sooo much everyone, once again u all saved me!