Question

Cos^2x+cos^2y-cos(x+y)cos(x-y)

Answer 1

$cos^2x+cos^2y-cos(x+y)*cos(x-y)= \\ =cos^2x+cos^2y-0.5(cos2y+cos2x)= \\ =cos^2x+cos^2y-0.5(2cos^2y-1+2cos^2x-1)= \\ =cos^2x+cos^2y-cos^2y-cos^2x-1=-1$

Answer 2

To simplify the expression $\cos^2x + \cos^2y - \cos(x+y)\cos(x-y)$, you can use some trigonometric identities. Here's how you can do it step by step:

1. Start with the expression:

$\cos^2x + \cos^2y - \cos(x+y)\cos(x-y)$

2. Recall the trigonometric identity $\cos(2\theta) = \cos^2\theta - \sin^2\theta$.

3. Rewrite $\cos^2x$ and $\cos^2y$ using this identity:

$\cos^2x = \frac{1}{2}(1 + \cos(2x))$
$\cos^2y = \frac{1}{2}(1 + \cos(2y))$

4. Substitute these expressions back into the original expression:

$\frac{1}{2}(1 + \cos(2x)) + \frac{1}{2}(1 + \cos(2y)) - \cos(x+y)\cos(x-y)$

5. Now, let's work on the last term $\cos(x+y)\cos(x-y)$. Use the trigonometric identity $\cos(A + B)\cos(A - B) = \frac{1}{2}[\cos(2A) + \cos(2B)]$:

$\cos(x+y)\cos(x-y) = \frac{1}{2}[\cos(2x) + \cos(2y)]$

6. Substitute this back into the expression:

$\frac{1}{2}(1 + \cos(2x)) + \frac{1}{2}(1 + \cos(2y)) - \frac{1}{2}[\cos(2x) + \cos(2y)]$

7. Now, combine like terms:

$\frac{1}{2} + \frac{1}{2} - \frac{1}{2}\cos(2x) + \frac{1}{2} - \frac{1}{2}\cos(2y) - \frac{1}{2}[\cos(2x) + \cos(2y)]$

8. Simplify further:

$\frac{1}{2} + \frac{1}{2} - \frac{1}{2}\cos(2x) + \frac{1}{2} - \frac{1}{2}\cos(2y) - \frac{1}{2}\cos(2x) - \frac{1}{2}\cos(2y)$

9. Combine like terms again:

$1 - \frac{1}{2}\cos(2x) - \frac{1}{2}\cos(2y) - \frac{1}{2}\cos(2x) - \frac{1}{2}\cos(2y)$

10. Combine the $\cos(2x)$ and $\cos(2y)$