# Prove Trig Identity

1. Aug 3, 2004

### Oxymoron

How would you prove using the cancellation laws 2arccos(x) = arccos(2x&sup2; - 1). I am stumped. Any guidance is appreciated.

2. Aug 3, 2004

### uart

Start by taking cos of both sides to get :

cos ( acos(x) + acos(x) ) = 2x^2 -1

It's pretty easy from there, so I'll leave itfor you to have a go. :)

3. Aug 3, 2004

### Oxymoron

What if I let &theta; = acos(x) which implies that x = cos&theta;

Then the expression becomes...

2&theta; = acos(2(cos&theta; )&sup2; - 1)
2&theta; = acos(2cos&sup2;&theta; - 1)

Then take cos of both sides and use the cancellation law...

cos2&theta; = 2cos&sup2;&theta; - 1

This is a well known trigonometric identity. So if we start from this and work backwards the identity in the question can easily be shown.

4. Aug 3, 2004

### Oxymoron

cos(2&theta;) = 2cos&sup2;&theta; -1

Then take the square outside the brackets

cos(2&theta;) = 2(cos&theta; )&sup2; -1

Since cos(acos(x)) = x we may write it as

cos(2&theta;) = cos(acos(2(cos&theta; )&sup2; -1))

Now take inverse cos of both sides

acos(cos(2&theta;)) = acos(cos(acos(2(cos&theta; )&sup2; -1)))

Cancel out the acos(cos(x)) terms

2&theta; = acos(2(cos&theta; )&sup2; -1)

Let &theta; = acos(&alpha;) implies &alpha; = cos&theta;

2acos(&alpha; ) = acos(2&alpha;&sup2; - 1)

QED

5. Aug 4, 2004

### uart

I just used the usual cos(A+B) expansion on the LHS to give :

cos^2( acos(x) ) - sin^2 (acos(x))

and then added zero in the form of cos^2(.) + sin^2(.) - 1 to the LHS to give,

2 cos^2( acos(x) ) - 1

6. Aug 5, 2004

### Oxymoron

Another question.

For what values of A is the trigonometric identity cos2A = 2cos²A - 1 valid? I thought it valid for all real numbers. But there must be a trick??

Any hints would be appreciated. Thanks.

7. Aug 5, 2004

### jtolliver

It is valid for all A. It follows from the pythagorean identity and the identity cos (A+B) = cos A cos B - sin A sin B. That identity gives cos 2A = cos (A+A) = cos² A - sin² A. Adding on cos² A + sin² A - 1 (which is 0 by the pythagorean identity) to the right side gives the identity cos 2A = 2cos² A - 1