Hard OCR CORE 4 Proof

Hi guys!

Okay I can't do this question, and have spent the last 2 hours on it:/

1a) By first expressing Cos4x in terms of Cos2x, show that Cos4x=8cos^4 - 8 cos^2 +1

And hence show that 8 Cos^4x= Cos4x + 4Cos2x +3.

I don't know where to start tbh!
I tried using Cos(2x+2x) to get Cos2x.Cos2x - Sin2x.Sin2x
Therefore I get Cos4x- Sin4x... now I have no idea what do :frown:

Help guys please!

Reply 1

BJack

19

\cos 2x \cos 2x \equiv \cos^2 2x \not\equiv \cos 4x

But you've got the right idea. Use Pythagoras to get everything in terms of cos:

\cos 4x \equiv \cos^2 2x -\sin^2 2x \equiv \cos^2 2x - (1- \cos^2 2x)

etc.

(edited 13 years ago)

Reply 2

Farhan.Hanif93

18

Original post by J DOT A

Hi guys!

Okay I can't do this question, and have spent the last 2 hours on it:/

1a) By first expressing Cos4x in terms of Cos2x, show that Cos4x=8cos^4 - 8 cos^2 +1

And hence show that 8 Cos^4x= Cos4x + 4Cos2x +3.

I don't know where to start tbh!
I tried using Cos(2x+2x) to get Cos2x.Cos2x - Sin2x.Sin2x
Therefore I get Cos4x- Sin4x... now I have no idea what do :frown:

Help guys please!

The bolded line is a mistake.

\cos (2x) . \cos (2x) - \sin (2x) . \sin (2x) \equiv \cos ^2(2x) - \sin ^2(2x) \not\equiv \cos (4x) - \sin (4x)

.

Can you recall an identity that connects

\sin ^2(2x)

and

\cos ^2(2x)

?

Spoiler

.

Then recall the expansion of cos2x and use the identity linking

\sin ^2 x

and

\cos ^2x

to get it all in terms of

\cos x

.

For the next part, note that:

\cos (4x)= 8\cos ^4(x) - 8 \cos ^2(x) + 1

\Leftrightarrow \cos (4x) = 8\cos ^4(x) - 8\cos ^2(x) +4 -3

.

Can you see the expansion of 4cos 2x in there.

Reply 3

Farhan.Hanif93

18

Original post by BJack

\cos^2 2x \neq \cos 4x

Are you sure about that, what about x=0? :wink:

You really need to use "equivalent to" signs

(\equiv)

rather than "equal to". They have a subtle difference in meaning.

Reply 4

J DOT A

OP

Original post by Farhan.Hanif93

Are you sure about that, what about x=0? :wink:

You really need to use "equivalent to" signs

(\equiv)

rather than "equal to". They have a subtle difference in meaning.

Ahh thank you! I got the answer!
However, I don't know how to show that 8Cos^4x= Cos4x + 4cos2x +3

So rearranging you get 8Cos^4x = Cos4x + 8Cos^2x -1

Now I don't know what to do!

(edited 13 years ago)

Reply 5

BJack

19

Original post by Farhan.Hanif93

Are you sure about that, what about x=0? :wink:

You really need to use "equivalent to" signs