Prove or disprove the following Identities:

cos(-x) - sin(-x) = cos(x) + sin (x)

sin raised to the 4 (theta) - cos raised to the 4 (theta) = sin squared (theta) - cos squared (theta)

cos (x+(pi)/(6)) + sin (x - (pi)/(3)) = 0

cos(x+y)cos(x-y) = cos squared (x) - sin squared (y)

Sorry if you don't understand anything

For the first problem:

cos(-x) - sin(-x) = cos(x) + sin(x)

A few identities for negatives:

cos(-x) = cos(x)
sin(-x) = -sin(x)

Therefore:

cos(x) - [-sin(x)] = cos(x) + sin(x)

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For your last problem:

cos(x+y)cos(x-y) = cos^2(x) - sin^2(y)

Some identities:
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

Multiplying using both identities:

[cos^2(x) cos^2(y)] - [sin^2(x) sin^2(y)]

Next, use the identity:
sin^2(x) + cos^2(x) = 1

[1-sin^2(x)][1-sin^2(y)] - [sin^2(x) sin^2(y)]

Multiply [1-sin^2(x)][1-sin^2(y)]:

1 - sin^2(y) - sin^2(x) + [sin^2(x) sin^2(y)] - [sin^2(x) sin^2(y)]

We are left with this:
1 - sin^2(y) - sin^2(x)

Which equals this:
cos^2(x) - sin^2(y)

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I'll stop there. I hope this helps.

No problem! Let's go through each identity one by one, explaining how to prove or disprove them.

Identity 1: cos(-x) - sin(-x) = cos(x) + sin(x)

To prove or disprove this identity, we can use the fact that cos(-x) = cos(x) and sin(-x) = -sin(x).

Substituting these values into the identity, we have:

cos(x) - (-sin(x)) = cos(x) + sin(x)

Simplifying both sides, we get:

cos(x) + sin(x) = cos(x) + sin(x)

Since both sides are equal, the identity is proved to be true.

Identity 2: sin^4(theta) - cos^4(theta) = sin^2(theta) - cos^2(theta)

To prove or disprove this identity, we can expand both sides and simplify.

Expanding sin^4(theta), we get:

(sin^2(theta))^2 = (sin^2(theta))(sin^2(theta))

Similarly, expanding cos^4(theta), we have:

(cos^2(theta))^2 = (cos^2(theta))(cos^2(theta))

So the left-hand side of the identity becomes:

(sin^2(theta))(sin^2(theta)) - (cos^2(theta))(cos^2(theta))

Expanding the right-hand side gives:

sin^2(theta) - cos^2(theta)

Now we can see that the left-hand side does not equal the right-hand side. Therefore, the identity is disproved.

Identity 3: cos(x+(pi)/6) + sin(x - (pi)/3) = 0

To prove or disprove this identity, we need to rewrite the trigonometric functions using their sum and difference formulas.

Using the formula for cos(x+y), we have:

cos(x+(pi)/6) = cos(x)cos((pi)/6) - sin(x)sin((pi)/6)

And using the formula for sin(x-y), we get:

sin(x - (pi)/3) = sin(x)cos((pi)/3) - cos(x)sin((pi)/3)

Substituting these expressions back into the original identity gives:

cos(x)cos((pi)/6) - sin(x)sin((pi)/6) + sin(x)cos((pi)/3) - cos(x)sin((pi)/3) = 0

Next, we simplify and combine like terms:

cos(x)(cos((pi)/6) - sin((pi)/3)) + sin(x)(cos((pi)/3) - sin((pi)/6)) = 0

Since this equation does not hold for all values of x, the identity is disproved.

Identity 4: cos(x+y)cos(x-y) = cos^2(x) - sin^2(y)

To prove or disprove this identity, we can use the formula for cos(x+y) and cos(x-y) and simplify both sides.

Using the formula for cos(x+y), we have:

cos(x+y) = cos(x)cos(y) - sin(x)sin(y)

Similarly, using the formula for cos(x-y), we get:

cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

Substituting these expressions back into the identity gives:

(cos(x)cos(y) - sin(x)sin(y))(cos(x)cos(y) + sin(x)sin(y)) = cos^2(x) - sin^2(y)

Simplifying both sides of the equation, we have:

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

Rearranging, we get:

cos^2(x)cos^2(y) + sin^2(y) - sin^2(x)sin^2(y) - cos^2(x) = 0

This equation does not hold for all values of x and y, so the identity is disproved.

Therefore, out of the four identities given, only the first one is proved to be true. The second, third, and fourth identities are disproved.