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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,051

Prove that

*Last edited by niharika_kumar (2014-04-17 04:17:45)*

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

hi Niharika,

No one else is offering a method so here is one that will work. It's not nice and 'slick' so hopefully someone else will jump in with a simpler method.

If you expand this you'll get the powers of 10 that you need (plus some unhelpful other powers of sin and cos)

But you can write all the cos^2 terms as (1 - sin^2) instead so the unhelpful terms will all become powers of sine.

Do the same for

Now use these formulas to construct the expression

It will come to 1 without all those powers of sine, so they must all cancel out. Job done!

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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Those expressions are factorable...

*Last edited by Agnishom (2014-04-17 14:30:42)*

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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Let cos^2 θ = y, sin^2 θ = x

We want to prove that 6(x^5 + y^5) - 15(x^4 + y^4) + 10(x^3 + y^3) - 1 = 0

Now, since cos^2 θ + sin^2 θ = 1, y=1-x

6(x^5 + y^5) = 6(x^5 + (1-x)^5) = 6 - 30 x + 60 x^2 - 60 x^3 + 30 x^4

-15(x^4 + y^4) = -15(x^4 + (1-x)^4) = -15 + 60 x - 90 x^2 + 60 x^3 - 30 x^4

10(x^3 + y^3) = 10(x^3 + (1-x)^3) = 10 - 30 x + 30 x^2

Adding up,

LHS = 6(x^5 + y^5) - 15(x^4 + y^4) + 10(x^3 + y^3) - 1 = 6 - 30 x + 60 x^2 - 60 x^3 + 30 x^4 -15 + 60 x - 90 x^2 + 60 x^3 - 30 x^4 + 10 - 30 x + 30 x^2 - 1 = 0 = RHS

##QED##

Sorry Niharka, we are still needing the binomial theorem. The above is very tedious to d by hand. I used a computer

*Last edited by Agnishom (2014-04-17 14:58:45)*

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,051

let me go and do some problems on binomial theorem first.

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Are you enjoying school?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,051

school hasn't started yet.

It will start only after the results come.

what about you?

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We will start this fridday.

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,051

great.

you are choosing PCM, right?along with which subject?

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Along with biology.

What about you? Computer Science?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,051

my mother wanted me to take PCMB and become a doctor(as I also scored highest everytime in biology), but i am very sensitive to things , so I dropped the idea.

I was confused between Comp. sc., Economics and P.E., then a comp.sc. student told me that you can learn programming anytime anywhere, so why sit for exams for that as even a small silly error in writing programs can cost loosing good marks.

So now I am confused between Economics and P.E.?

What do you suggest?

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