Proving equations using sin, cos and tan: (1+tanx)/(sinx+cosx) = 1/cosx

[paleksandra]

Hello there!

I am studying high school mathematics and am having trouble solving a triginometry problem where I have two mathematical statements with an equal sign between (ex. ab = cd). The assignment is to prove that ab = cd and I have gotten a few steps but get stuck in the middle.

This is the statement:

(1+tanx)/(sinx+cosx) = 1/cosx

In other words I have to show that these two statements equal each other.

I have this so far:

(1+tanx)/(sinx+cosx) = 1/cosx

[1+(sinx/cosx)]/sinx+cosx = 1/cosx (as I know that tanx = sinx/cosx)

I know that I somehow have to multiply the left hand side by sinx+cosx (according to my book) but I don't know how to do this.

Perhaps if someone shows me how to do this next step I could figure out the rest.

Thanks!

 

[pka]

This is the statement:
(1+tanx)/(sinx+cosx) = 1/cosx

First of all please use proper function notation. It should be \(\displaystyle \cos(x)\) not \(\displaystyle \cosx\).

\(\displaystyle \dfrac{1+\frac{\sin(x)}{\cos(x)}}{\sin(x)+\cos(x)}=\dfrac{\cos(x)+\sin(x)}{(\cos(x))(\sin(x)+\cos(x))}\)

 

[paleksandra]

First of all please use proper function notation. It should be \(\displaystyle \cos(x)\) not \(\displaystyle \cosx\).

\(\displaystyle \dfrac{1+\frac{\sin(x)}{\cos(x)}}{\sin(x)+\cos(x)}=\dfrac{\cos(x)+\sin(x)}{(\cos(x))(\sin(x)+\cos(x))}\)

I did not mean to offend anyone with my function notation. In my mathematics book it is written in exactly the way that I have written it here so I can only assume that it is also a correct notation since the book is written by a mathematician.

Also, I still don't understand how to get from:

[1+(sinx/cosx)]/sinx+cosx to 1/cosx

Could someone show me the steps in between, how to write it out step by step?

Thanks!

 

[stapel]

This is the statement:

(1+tanx)/(sinx+cosx) = 1/cosx

In other words I have to show that these two statements equal each other.

I have this so far:

(1+tanx)/(sinx+cosx) = 1/cosx

[1+(sinx/cosx)]/sinx+cosx = 1/cosx (as I know that tanx = sinx/cosx)

What did you get when you converted the "1" on the left-hand side to the common denominator with the fraction in the numerator? What common factor did you then have? What happened when you cancelled this off?

 

[Deleted member 4993]

I did not mean to offend anyone with my function notation. In my mathematics book it is written in exactly the way that I have written it here so I can only assume that it is also a correct notation since the book is written by a mathematician.

Also, I still don't understand how to get from:

[1+(sinx/cosx)]/sinx+cosx to 1/cosx

Could someone show me the steps in between, how to write it out step by step?

Thanks!

That probably is not true. If the () is skipped - generally a space is inserted. So your book writes sin x, cos x, etc. (notice the space between sin and x).

So your book (written by a mathematician) may have correct notation - but you have failed to reproduce it.

Regarding the problem in hand, did you notice the next step already provided by the poster?