Are the Functions sin^2(x), cos^2(x), and cos(2x) Linearly Independent?

[Naeem]

Q. Prove whether or not the following functions are linearly independent.

1. sin^2x ( "sine squared x")

2. cos^2x ("cosine squared x")

3. cos2x

Please any help is greatly appreciated on these parts:

 

[Hurkyl]

Please read the guidelines for posting homework help questions.

https://www.physicsforums.com/showthread.php?t=4825

You must have had a thought on this problem already -- surely you know, say, the definition, or a relevant theorem?


P.S. it's customary to put some sort of punctuation between a trig function and its argument. You probably wanted to say cos 2x or cos(2x), not cos2x, and cos^2 x, not cos^2x.

 

[thepatientmental]

To prove whether or not the given functions are linearly independent, we need to show that no linear combination of them can equal to the zero function, except for the trivial combination where all coefficients are zero.

First, let's consider the linear combination of these functions:

a*sin^2x + b*cos^2x + c*cos2x = 0

We can rewrite this as:

a*(1-cos^2x) + b*cos^2x + c*cos2x = 0

Next, we can use the trigonometric identity cos^2x = 1 - sin^2x to simplify the equation:

a + (b-c)*cos^2x + c*cos2x = 0

Now, we can see that in order for this equation to hold for all values of x, the coefficients must all be equal to zero. Therefore, a = b = c = 0, which is the trivial combination. This shows that the given functions are linearly independent.

To further prove this, we can also use the Wronskian determinant. The Wronskian of a set of functions is defined as:

W(f1, f2, ..., fn) = det([f1, f2, ..., fn; f1', f2', ..., fn'])

Where f1', f2', ..., fn' are the derivatives of the functions. If the Wronskian of a set of functions is non-zero, then the functions are linearly independent.

In this case, the Wronskian determinant is:

W(sin^2x, cos^2x, cos2x) = det([sin^2x, cos^2x, cos2x;2sinxcosx, -2sinxcosx, -2sin2x])

= -2sin2x(det([sin^2x, cos2x; 2sinxcosx, -2sin2x]))

= -2sin2x(-2sin^2x - 2cos^2x)

= 4sin2x(cos^2x - sin^2x)

= 4sin2x(cos2x)

Since cos2x is not equal to zero for all values of x, the Wronskian determinant is non-zero. Therefore, the given functions are linearly independent.

In conclusion, both methods of proof show that the functions sin^2x, cos^2