Need help on a grade 12 Adv. Func Trig. Identity

[Detestify]

Hi,

I need help with an identity for my grade 12 Advanced Functions class.

The identity is (Cosx-Sinx)/(Cosx+Sinx) = Sec2x-Tan2x

Here is what I have worked out so far.

RS = 1/Cos2x - Sin2x/Cos2x

= 1/Cos(x+x) - Sin(x+x)/Cos(x+x)

= 1/CosxCosx-SinxSinx - SinxCosx+CosxSinx/CosxCosx-SinxSinx

My problem is that I don't know how to proceed further than that. Any and all help is appreciated.

Thanks in advance!

 

[Detestify]

Ok, I'm gonna need you to put parentheses around the bits that need to be grouped together so we make sure we are both talking about the same thing.

I think you mean for your identity

\(\displaystyle \frac{cosx-sinx}{cosx+sinx}=sec2x-tan2x\)

Is that correct?

Ok I see that it is.

There's no particular magic to this problem. You just have to carefully expand everything out. I guess there's one small trick regarding the left hand side.

try multiplying the left hand side top and bottom by \(\displaystyle (cosx-sinx)\)

then note that \(\displaystyle cos2x=(cos^2x-sin^2x)\)

and that \(\displaystyle sin2x=2sinx\cdot cosx\)

see if you can work it now.

How do I justify just multiplying the LS by (cosx-sinx)?

 

[Detestify]

top and bottom,

multiply it by 1 in the form of \(\displaystyle \frac{cosx-sinx}{cosx-sinx}\)

Here's where I am at now.

RS = 1/Cos2x - Sin2x/Cos2x

= 1/Cos(x+x) - Sin(x+x)/Cos(x+x)

= 1/CosxCosx-SinxSinx - SinxCosx+CosxSinx/CosxCosx-SinxSinx

= 1/Cos^2x-Sin^2x - 2sinxcosx/Cos^2x-Sin^2x

= 1-2sinxcosx/Cos^2x-Sin^2x

LS = 1 . Cosx-Sinx/Cosx+Sinx

= Cosx-sinx/cosx-sinx . Cosx-sinx/cosx+sinx

 

[Detestify]

I hate to be picky but I'm gonna have to ask you to use parentheses to group things properly otherwise I really don't know exactly what your equations are.

No worries.

RS = 1/Cos2x - Sin2x/Cos2x

= 1/Cos(x+x) - Sin(x+x)/Cos(x+x)

= 1/(CosxCosx-SinxSinx) - (SinxCosx+CosxSinx)/(CosxCosx-SinxSinx)

= 1/(Cos^2x-Sin^2x) - (2sinxcosx)/(Cos^2x-Sin^2x)

= (1 - 2sinxcosx)/(Cos^2x-Sin^2x)

LS = 1 . (Cosx-Sinx)/(Cosx+Sinx)

= (Cosx-sinx)/(cosx-sinx) . (Cosx-sinx)/(cosx+sinx)

Is that better?

The . is multiplication.

 

[Detestify]

yes that's good. don't stop with the left hand side, keep going. You can expand all that out. Things will start to magically cancel with the right hand side. Persevere!

Got it. Thanks a bunch Romsek!