anonymous
  • anonymous
Verify the identity. cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
\[\cot^2x/\csc x +1 = 1 - \sin x/\sin x\]
welshfella
  • welshfella
try changing LHS to sine and cosines
anonymous
  • anonymous
(cos^2(x)/sin^2(x))/1/sin(x) = (cos^2(x)/sin^2(x))/sin(x)

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welshfella
  • welshfella
just checking is the question |dw:1438357529236:dw|
anonymous
  • anonymous
(cos^2(x)/sin^2(x))/1/sin(x)+sin(x)/sin(x) = (cos^2(x)/sin^2(x))/sin(x)
anonymous
  • anonymous
yes
welshfella
  • welshfella
ok thnx
welshfella
  • welshfella
i'll write s for cos x and s for sin x |dw:1438357934242:dw| --
welshfella
  • welshfella
can you continue? 2 more steps and you are home
anonymous
  • anonymous
almost let me look at it
welshfella
  • welshfella
(1 - s) will cancel out
welshfella
  • welshfella
sorry (1 + s) will cancel out
anonymous
  • anonymous
on the third step where did you get c^2/s^2
welshfella
  • welshfella
c^2 / s^2 was carried on from second step
anonymous
  • anonymous
so the second step was simplified to that
welshfella
  • welshfella
- not very readable I know!
welshfella
  • welshfella
yes
anonymous
  • anonymous
ok so 1+s/s simplifies to s^2?
welshfella
  • welshfella
no in the second step c^2 / s^2 is divided by (1+s) / s so we invert the (1+ s) / s to s/(1 + s) then multiply
anonymous
  • anonymous
I understand now I just had to look at the equation a bit more
welshfella
  • welshfella
and in 4th step we replace cos^x by 1 - sin^ x
welshfella
  • welshfella
cos^2 x by 1 - sin^2 x
anonymous
  • anonymous
ok
welshfella
  • welshfella
this is the difference of 2 squares so simplifies to (1 - s)(1 + s)
anonymous
  • anonymous
yep i understand that
anonymous
  • anonymous
im ready for the next two steps
welshfella
  • welshfella
so now we can cancel out (1 + s) which appears in the top and bottom of the fractions |dw:1438358863645:dw|
anonymous
  • anonymous
thanks so much
welshfella
  • welshfella
yw
anonymous
  • anonymous
\[\frac{1-\sin(x)}{\sin(x)}=\frac{1}{\sin(x)}-\frac{\sin(x)}{\sin(x)}=\csc(x)-1\] \[\implies (\csc(x)-1).\frac{(\csc(x)+1)}{(\csc(x)+1)}=\frac{\csc^2(x)-1}{\csc(x)+1}=\frac{\cot^2(x)}{\csc(x)+1}\]
anonymous
  • anonymous
I've made use of the identities \[(a-b)(a+b)=a^2-b^2\] and \[1+\cot^2(x)= \csc^2(x)\]

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