anonymous
  • anonymous
cot(theta)-tan(theta) over sin(theta) + cos(theta) equals csc(theta) - sec(theta) how do I proof this?
TriC-MathMOOC
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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Mertsj
  • Mertsj
Let's use x instead of theta
anonymous
  • anonymous
okay
Mertsj
  • Mertsj
\[\frac{\cot x-\tan x}{\sin x+\cos x}=\csc x-\sec x\]

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Mertsj
  • Mertsj
Do I have the problem stated correctly?
anonymous
  • anonymous
yes :)
anonymous
  • anonymous
would I have to replace cot(x) with cos(x)/sin(x)
anonymous
  • anonymous
and tan(x) with sin(x)/cos(x)
Mertsj
  • Mertsj
\[(\frac{\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}}{\sin x+\cos x})\times \frac{\sin x \cos x}{\sin x \cos x}=\frac{\cos ^2x-\sin ^2x}{\sin x \cos x(\sin x+\cos x)}\]
Mertsj
  • Mertsj
Can you get it from there?
anonymous
  • anonymous
sadly no? can you explain how I can simplify it more please?
Mertsj
  • Mertsj
Factor the numerator and then the cosx + sinx will cancel.
anonymous
  • anonymous
I'm still lost :(
Mertsj
  • Mertsj
@alejandrop95 This is not your question
anonymous
  • anonymous
how would I factor it though? I'm sorry I am new to trig.
Mertsj
  • Mertsj
Can you factor this: \[a^2-b^2\]
anonymous
  • anonymous
I don't think you can because it doesn't have the same coefficient
Mertsj
  • Mertsj
\[a^2−b^2=(a-b)(a+b)\]
Mertsj
  • Mertsj
\[\cos ^2x-\sin ^2x=(\cos x-\sin x)(\cos x+\sin x)\]
anonymous
  • anonymous
oh okay that makes sense. so would it be cos-sin(cos+sin)
Mertsj
  • Mertsj
yes
anonymous
  • anonymous
then that would cancel with the denominator correct?
Mertsj
  • Mertsj
Yes. sinx + cosx will cancel
anonymous
  • anonymous
which will leave 1/sin(x)+cos(x)
Mertsj
  • Mertsj
no
anonymous
  • anonymous
using the reciprocal identities this will equal csc(x)-sec(x) ?
Mertsj
  • Mertsj
|dw:1382579245755:dw|
anonymous
  • anonymous
cos(x)-sin(x)/sin(x)+cos(x0
anonymous
  • anonymous
this is where I would use the reciprocal identities?
Mertsj
  • Mertsj
|dw:1382579370642:dw|
anonymous
  • anonymous
the cos(x) cancel and the sin(x) cancels?
Mertsj
  • Mertsj
|dw:1382579509266:dw|
anonymous
  • anonymous
yay!!!! the cancel and its one,which now you are left with the csc(x) and sec(x) reciprocal identities!! Thank you!! the key idea here that I need to work on was the factoring
anonymous
  • anonymous
thank you !:)
anonymous
  • anonymous
thanks ! :)
Mertsj
  • Mertsj
yw

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