Solving Identities Homework: Proving 1/1+sinx=sec2x - tanx/cosx

[ritagogna]

Homework Statement

hey i really need some ones help i need to prove these 2 identities. I am in grade 12 and I've been working on these questions for almost 2 hours and keep failing pleasezz help
b. 1/1+sinx = sec2x - tanx/cosx

Homework Equations

-secx=1/cosx
-tanx=sinx/cosx
-sin2x+cos2x=1
-cscx=1/sinx

The Attempt at a Solution

So i tried proving the identity and got stuck...
i picked one side which was 1/1+sinx
then i got the conjugate
=1/1+sinx(1-sinx/1-sinx)
=1-sinx(1/sinx)
and this is where i have no idea what to do because i need to make it equal to sec2x-tanx/cosx

 

[cristo]

By sec2x do you mean sec²x? If so, the equation is $$\frac{1}{1+sinx}=sec^2x-\frac{tanx}{cosx}$$.

Now, my advice would be to write out tanx and secx in terms of cosx and sinx. Then multiply the equation to get rid of the denominators. Try this and post what you get.

 

[quantumdude]

=1/1+sinx(1-sinx/1-sinx)

I take it that you mean:

$$\frac{1}{1+\sin(x)}\frac{1-\sin(x)}{1-\sin(x)}$$

=1-sinx(1/sinx)

I'm not sure of how you got this (or even of how to read it!), but it definitely looks wrong. You should multiply across the top and across the bottom in my expression above. Try that, and post what you get.

 

[ritagogna]

so i worked it out and I am pretty sure its 1- sinx/ cosx

 

[quantumdude]

Nope, the cosine should be squared.

 

[ritagogna]

no the equation is exacly what cristo got

 

[ritagogna]

ok so its 1-sinx over cos2x

 

[quantumdude]

no the equation is exacly what cristo got

No, I'm talking about the part of your solution that I quoted.

 

[ritagogna]

wait no its 1/cos2x-sinx/cosx

 

[quantumdude]

ok so its 1-sinx over cos2x

Do you mean cos²(x)? If so, then you're right. But if you mean cos(2x), then you're not right.

 

[ritagogna]

no i mean cos squared 2 x but i still haven't solved the identity and i don't know where to go from here because ill just be making the equation more complicated by converting the identities

 

[quantumdude]

no i mean cos squared 2 x

That's wrong. When you multiply $1+\sin(x)$ by $1-\sin(x)$, you get $1-\sin^2(x)=\cos^2(x)$. The argument of the cosine is not $2x$.