# Prove a Trigonometry identity

Discussion in 'General Off-Topic Chat' started by shakirmoledina, Dec 1, 2012.

# Prove a Trigonometry identity

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1. ### shakirmoledinaLegend

Member
4
Oct 23, 2004
Dar es Salaam
Hey guys,
For those who are fans of pure maths, help me prove the following identity

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

I tested it through a calculator and they match so its only the theoretical matching that I cannot do.

things you can use are:
sin 2 x + cos 2 x = 1
and tanx = sinx/cosx

2. ### VehoThe man who cried "Ni".

Former Staff
16
Apr 4, 2006
Zagreb
Oh all right. One clarification needed, is the (1-sinx) part of the denominator, or is the whole fraction (cosx/tanx) multiplied by it?

Here you go:

cosx/[(sinx(1-sinx))/cosx] = sinx/sinx + 1/sinx

cos^2(x)/[sinx(1-sinx)] = (1+sinx)/sinx ///multiply all by sinx(1-sinx)

cos^2x = (1+sinx)(1-sinx) = 1 - sin^2x.

Which is true.

3. ### shakirmoledinaLegend

Member
4
Oct 23, 2004
Dar es Salaam
uhuh so can u prove it from one side only so that the multiplication of the denominator doesnt come into place?
Its a different approach where u consider both sides at once.

Its for a boy who has his exams on monday.

4. ### VehoThe man who cried "Ni".

Former Staff
16
Apr 4, 2006
Zagreb

cosx/[(sinx(1-sinx))/cosx] = cos^2x/[sinx(1-sinx)] =

= (1-sin^2x)/[sinx(1-sinx)] =///using the equivalence that a^2-b^2 = (a+b)(a-b) /// =

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

=(1+sinx)/sinx =

= 1/sinx + sinx/sinx =

= 1 + 1/sinx

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5. ### shakirmoledinaLegend

Member
4
Oct 23, 2004
Dar es Salaam
equivalence... awesome veho
vehi awesome

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13. ### leic7GBAtemp Regular

Member
3
Dec 22, 2006
Another strategy for proving something like that is to separate all the 'complex' looking fractions to ONE side of the equation, so that the other side contains only a 'simple' term. In general, it's easier to reduce a complex formula to something simpler, than it's to try and match one complex formula with another complex formula. so for example, move the last term 1/sinx to the left side, and solve for
cosx/[tanx(1-sinx)] - 1/sinx

If you get 1, you're done.

14. ### shakirmoledinaLegend

Member
4
Oct 23, 2004
Dar es Salaam
YOU might wanna decomplexify your statement there. no full stop in that sentence.

15. ### leic7GBAtemp Regular

Member
3
Dec 22, 2006
Sorry, I just meant that for an equation such as:

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

...there are fractions on both sides of "=". Here, both "cosx/tanx(1-sinx)" and "1/sinx" are fractions.

If you can isolate these fractions to the left side of "=", the right side of "=" is left with the integer "1":

cosx/tanx(1-sinx) = 1 + 1/sinx (1)
cosx/tanx(1-sinx) - 1/sinx = 1 (2)

To prove (1) is equivalent to proving (2), to prove (2) you can simply reduce "cosx/tanx(1-sinx) - 1/sinx" to "1". I think a student would find it relatively easier if they know the "answer" is "1", and all they have to do is "solve" it.

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16. ### shakirmoledinaLegend

Member
4
Oct 23, 2004
Dar es Salaam
uhuh i like that since thats exactly the things that are required for solving such questions