How Do You Simplify the Expression sec + csc / (1 + tan)?

[PHK]

Homework Statement

sec+csc/1+tan

The Attempt at a Solution

i tried simplifying it and the farthest i got was: 1/cos + 2/sin + cos/sin^2
im not sure that's even right

 

[rocomath]

ok I'm a bit confused on what your problem is ... please make it a lil more clear

 

[PHK]

i have to simplify sec+csc/1+tan

 

[rocomath]

the problem is ...

$$\sec{x}+\frac{\csc{x}}{1}+\tan{x}$$

why did you write the 1 ... ?

 

[PHK]

its not that its, sec+csc and all that over 1+tan. sec+csc
.............1+tanignore the the dots the problem i am and talking about is in the the top right

also (sec+csc)/(1+tan)

 

[cristo]

You should use parentheses to make your questions unambiguous in future. Also, note that sec doesn't mean anything; you mean sec(x). Ok, well you could write everything in terms of sines and cosines first. Just substitute into that expression, then try and tidy up.

 

[rocomath]

(sec+csc)/(1+tan) ... much clearer

hopefully you see why i was confused, if you want help you got to be clear! or we're going to be going back and forth w/o you ever receiving help. lol

 

[dynamicsolo]

its not that its, sec+csc and all that over 1+tan. sec+csc
.............1+tan


ignore the the dots the problem i am and talking about is in the the top right

also (sec+csc)/(1+tan)

How are each of these trig functions defined? Replace each one by its definition and see what you get when you add the appropriate fractions.

 

[PHK]

yea sorry for not being clear.

and i already tryed replacing them by the definition. i just end up with 1/cos + 2/sin + cos/sin^2

 

[rocomath]

what did you multiply by?

i multiplied both NUM and DEN by cosx

i don't really see what else can be done.

 

[dynamicsolo]

yea sorry for not being clear.

and i already tryed replacing them by the definition. i just end up with 1/cos + 2/sin + cos/sin^2

I don't see how you got 2/sin, but I can tell you that this simplifies a lot.

Just make the replacements first, then add the fractions in the numerator and in the denominator separately. You'll have a compound fraction, where you can then invert the denominator and multiply. You'll see a couple of things that you can then cancel.

[ (1/cos) + ? ] / [ 1 + ? ] = ?

 

[PHK]

maybe that's wrong then (1/cos + 2/sin + cos/sin^2), also i tried multiplying by cosx and i get 1/sin + cos - 1/cos i tried going further but it seems like I am doing something wrong. does anyone have the solution yet?

 

[EugP]

$$\frac{\sec + \csc}{1+\tan}=\frac{\frac{1}{\cos} + \frac{1}{\sin}}{1+\frac{\sin}{\cos}}=\frac{\frac{\sin + \cos}{\cos \sin}}{\frac{\cos+\sin}{\cos}=\frac{1}{\sin}=\csc$$

 

[dynamicsolo]

maybe that's wrong then (1/cos + 2/sin + cos/sin^2), also i tried multiplying by cosx and i get 1/sin + cos - 1/cos i tried going further but it seems like I am doing something wrong. does anyone have the solution yet?

The next step should be

[ (1/cos) + (1/sin) ] / [ 1 + (sin/cos) ] . Add the fractions in the numerator and in the denominator, then simplify the compound fraction you get.

The answer is csc x .

 

[PHK]

[ (1/cos) + (1/sin) ] / [ 1 + (sin/cos) ] that's the original problem. how did you get csc from that?

 

[dynamicsolo]

$$\frac{\sec + \csc}{1+\tan}=\frac{\frac{1}{\cos} + \frac{1}{\sin}}{1+\frac{\sin}{\cos}}=\frac{\frac{\sin + \cos}{\cos \sin}}{\frac{\cos+\sin}{\cos}=\frac{1}{\sin}=\csc$$

Having looked at your source code, I think I should say that you should resist the temptation to provide the full solution to a problem. This forum is intended to guide students, rather than simply hand them the answers...

 

[dynamicsolo]

[ (1/cos) + (1/sin) ] / [ 1 + (sin/cos) ] that's the original problem. how did you get csc from that?

What does you get when you add (1/cos) + (1/sin) ? How about 1 + (sin/cos) ?

 

[PHK]

oh i get it thanks i was adding the top part properly but not hte bottm part. thanks