Integration (1 Viewer)

twinklegal19 · Oct 11, 2012

$\\$(i)if $y=\frac{1}{1+\tan{x}}$ express $\frac{dy}{dx}$ in terms of $\sin{2x}\\$(ii)By substituting $x=\frac{\pi}{4}-t$ show that $\int \frac{dx}{1+\sin{x}}=\frac{1}{2}\tan\left ( x-\frac{\pi}{4} \right )+C\\$(iii)Using the results of (i) and (ii) prove the identity $\frac{2}{1+\tan{x}}=1-\tan{\left ( x-\frac{\pi}{4} \right )}$

I'm having trouble with part (iii), any tips on how to prove it?

Thanks

edit: the answer to part (i) is

$-\frac{1}{1+\sin{2x}}$

SpiralFlex · Oct 11, 2012

Sorry on phone gonna take a while to type

barbernator · Oct 11, 2012

start with LHS.

get the 1 off the fraction.

Multiply top and bottom of fraction by cos

use the external angle method to get sin(x-pi/4) on top and cos(x-pi/4) on the bottom

Carrotsticks · Oct 11, 2012

What is t defined to be, in (i)? Some constant? t=tan(x/2)?

SpiralFlex · Oct 11, 2012

Carrotsticks said:
What is t defined to be, in (i)? Some constant? t=tan(x/2)?

This. But it makes sense if it is 1

Carrotsticks · Oct 11, 2012

SpiralFlex said:
This. But it makes sense if it is 1

$\\ $Yep, because if we consider $ t $ to be some arbitrary constant, then we acquire $ \\ \frac{dy}{dx} = - \frac{1}{\sin ^2 x + t^2 \cos ^2 x + 2t \sin x \cos x} \\\\ $And clearly when $ t=1 $ we have the required expression in terms of $ \sin(2x). $However, now part (ii) doesn't make sense.$$

twinklegal19 · Oct 11, 2012

Carrotsticks said:
What is t defined to be, in (i)? Some constant? t=tan(x/2)?

~~I'm not exactly sure, I copied the questions word for word so I don't know any more than you unfortunately. I treated it as a constant in (ii) though. I will type out my solution in a sec~~
EDIT: nvm, misread the question

SpiralFlex said:
Sorry on phone gonna take a while to type

Thanks for your effort

This question might look familiar in a certain *coughs*miscellaneous topics in calculus II*coughs* booklet

barbernator said:
start with LHS.

get the 1 off the fraction.

Multiply top and bottom of fraction by cos

use the external angle method to get sin(x-pi/4) on top and cos(x-pi/4) on the bottom

Thanks, will try that out

twinklegal19 · Oct 11, 2012

OMG I'm so sorry carrot and everyone else

that "t" in the first part (i) was meant to be a 1...

apologies for the confusion

Carrotsticks · Oct 11, 2012

twinklegal19 said:
OMG I'm so sorry carrot and everyone else

that "t" in the first part (i) was meant to be a 1...

apologies for the confusion

So what about the substitution in (ii), unless they want us to use the t-formulae?

twinklegal19 · Oct 11, 2012

Carrotsticks said:
So what about the substitution in (ii), unless they want us to use the t-formulae?

no t-formula is needed I think. I treated it as a normal variable and I still got the right answer

I could be wrong though

Integration (1 Viewer)

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