Let's examine each of the given trigonometric identities to determine which are true.
### Identity A
[tex]\[
\cos (x+y) + \cos (x-y) = 2 \cos x \cos y
\][/tex]
To verify this identity, we'll use the sum-to-product identities:
[tex]\[
\cos (x+y) + \cos (x-y) = 2 \cos \left(\frac{(x+y) + (x-y)}{2}\right) \cos \left(\frac{(x+y) - (x-y)}{2}\right)
\][/tex]
Simplifying each argument:
[tex]\[
\frac{(x+y) + (x-y)}{2} = \frac{2x}{2} = x
\][/tex]
[tex]\[
\frac{(x+y) - (x-y)}{2} = \frac{2y}{2} = y
\][/tex]
So we have:
[tex]\[
\cos (x+y) + \cos (x-y) = 2 \cos x \cos y
\][/tex]
Thus, identity A is true.
### Identity B
[tex]\[
\sin (x+y) + \sin (x-y) = 2 \cos x \sin y
\][/tex]
To verify this identity, we'll use the sum-to-product identities:
[tex]\[
\sin (x+y) + \sin (x-y) = 2 \sin \left(\frac{(x+y) + (x-y)}{2}\right) \cos \left(\frac{(x+y) - (x-y)}{2}\right)
\][/tex]
Simplifying each argument:
[tex]\[
\frac{(x+y) + (x-y)}{2} = \frac{2x}{2} = x
\][/tex]
[tex]\[
\frac{(x+y) - (x-y)}{2} = \frac{2y}{2} = y
\][/tex]
So we have:
[tex]\[
\sin (x+y) + \sin (x-y) = 2 \sin x \cos y
\][/tex]
Given identity states [tex]\( 2 \cos x \sin y \)[/tex] instead of [tex]\( 2 \sin x \cos y \)[/tex], thus this identity is false.
### Identity C
[tex]\[
\cos (x+y) - \cos (x-y)=2 \sin x \sin y
\][/tex]
To verify this identity, we'll use another sum-to-product identity:
[tex]\[
\cos (x+y) - \cos (x-y) = -2 \sin \left(\frac{(x+y) + (x-y)}{2}\right) \sin \left(\frac{(x+y) - (x-y)}{2}\right)
\][/tex]
Simplifying each argument:
[tex]\[
\frac{(x+y) + (x-y)}{2} = \frac{2x}{2} = x
\][/tex]
[tex]\[
\frac{(x+y) - (x-y)}{2} = \frac{2y}{2} = y
\][/tex]
So we have:
[tex]\[
\cos (x+y) - \cos (x-y) = -2 \sin x \sin y
\][/tex]
Given identity states [tex]\(2 \sin x \sin y \)[/tex] without the negative sign, so this identity is false.
### Identity D
[tex]\[
\tan (x - \pi) = \tan x
\][/tex]
To verify this, we use the periodic property of the tangent function. The tangent function has a period of [tex]\(\pi\)[/tex], which means:
[tex]\[
\tan (x - \pi) = \tan x
\][/tex]
This identity is true.
Therefore, the true identities are:
- A. [tex]\(\cos (x+y) + \cos (x-y) = 2 \cos x \cos y\)[/tex]
- D. [tex]\(\tan (x-\pi) = \tan x\)[/tex]
