Answer :
To determine which of the given expressions are identities, we'll examine each expression individually to see if it simplifies to a standard trigonometric identity.
### Expression A:
[tex]\[ \tan\left(x - \frac{\pi}{4}\right) = \tan x - 1 \][/tex]
To verify this, we check if both sides simplify to the same expression.
[tex]\[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - \tan\frac{\pi}{4}}{1 + \tan x \tan\frac{\pi}{4}} = \frac{\tan x - 1}{1 + \tan x \cdot 1} = \frac{\tan x - 1}{1 + \tan x} \][/tex]
Comparing this to [tex]\(\tan x - 1\)[/tex], it's clear that [tex]\(\frac{\tan x - 1}{1 + \tan x} \neq \tan x - 1\)[/tex]. Hence, this is not an identity.
### Expression B:
[tex]\[ \tan x - \tan y = \frac{\sin (x - y)}{\cos x \cos y} \][/tex]
We use the difference of tangents and angle difference formulas:
[tex]\[ \tan x - \tan y = \frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y} \][/tex]
[tex]\[ \frac{\sin (x - y)}{\cos x \cos y} = \frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y} \][/tex]
Both sides simplify to the same expression, confirming this is an identity.
### Expression C:
[tex]\[ \cos\left(x + \frac{\pi}{6}\right) = -\sin\left(x - \frac{\pi}{3}\right) \][/tex]
To verify this, let's use the angle addition and subtraction formulas:
[tex]\[ \cos\left(x + \frac{\pi}{6}\right) = \cos x \cos \frac{\pi}{6} - \sin x \sin \frac{\pi}{6} = \cos x \frac{\sqrt{3}}{2} - \sin x \frac{1}{2} \][/tex]
[tex]\[ -\sin\left(x - \frac{\pi}{3}\right) = -(\sin x \cos \frac{\pi}{3} - \cos x \sin \frac{\pi}{3}) = -(\sin x \frac{1}{2} - \cos x \frac{\sqrt{3}}{2}) = \cos x \frac{\sqrt{3}}{2} - \sin x \frac{1}{2} \][/tex]
Both sides simplify to the same expression, confirming this is an identity.
### Expression D:
[tex]\[ 1 - \tan x \tan y = \frac{\sin (x + y)}{\sin x \sin y} \][/tex]
To verify this, let's use the tangent and sine addition formulas:
[tex]\[ \tan x \tan y = \frac{\sin x \sin y}{\cos x \cos y} \][/tex]
[tex]\[ 1 - \tan x \tan y = 1 - \frac{\sin x \sin y}{\cos x \cos y} = \frac{\cos x \cos y - \sin x \sin y}{\cos x \cos y} \][/tex]
[tex]\[ \frac{\sin (x + y)}{\sin x \sin y} = \frac{\sin x \cos y + \cos x \sin y}{\sin x \sin y} \][/tex]
Since these expressions don't simplify to a common form between both sides, this is not an identity.
### Conclusion
From the above checks, the expressions that are identities are:
- Expression B: [tex]\(\tan x - \tan y = \frac{\sin (x - y)}{\cos x \cos y}\)[/tex]
- Expression C: [tex]\(\cos\left(x + \frac{\pi}{6}\right) = -\sin\left(x - \frac{\pi}{3}\right)\)[/tex]
Thus, the identities are [tex]\(\boxed{B \text{ and } C}\)[/tex].
### Expression A:
[tex]\[ \tan\left(x - \frac{\pi}{4}\right) = \tan x - 1 \][/tex]
To verify this, we check if both sides simplify to the same expression.
[tex]\[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - \tan\frac{\pi}{4}}{1 + \tan x \tan\frac{\pi}{4}} = \frac{\tan x - 1}{1 + \tan x \cdot 1} = \frac{\tan x - 1}{1 + \tan x} \][/tex]
Comparing this to [tex]\(\tan x - 1\)[/tex], it's clear that [tex]\(\frac{\tan x - 1}{1 + \tan x} \neq \tan x - 1\)[/tex]. Hence, this is not an identity.
### Expression B:
[tex]\[ \tan x - \tan y = \frac{\sin (x - y)}{\cos x \cos y} \][/tex]
We use the difference of tangents and angle difference formulas:
[tex]\[ \tan x - \tan y = \frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y} \][/tex]
[tex]\[ \frac{\sin (x - y)}{\cos x \cos y} = \frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y} \][/tex]
Both sides simplify to the same expression, confirming this is an identity.
### Expression C:
[tex]\[ \cos\left(x + \frac{\pi}{6}\right) = -\sin\left(x - \frac{\pi}{3}\right) \][/tex]
To verify this, let's use the angle addition and subtraction formulas:
[tex]\[ \cos\left(x + \frac{\pi}{6}\right) = \cos x \cos \frac{\pi}{6} - \sin x \sin \frac{\pi}{6} = \cos x \frac{\sqrt{3}}{2} - \sin x \frac{1}{2} \][/tex]
[tex]\[ -\sin\left(x - \frac{\pi}{3}\right) = -(\sin x \cos \frac{\pi}{3} - \cos x \sin \frac{\pi}{3}) = -(\sin x \frac{1}{2} - \cos x \frac{\sqrt{3}}{2}) = \cos x \frac{\sqrt{3}}{2} - \sin x \frac{1}{2} \][/tex]
Both sides simplify to the same expression, confirming this is an identity.
### Expression D:
[tex]\[ 1 - \tan x \tan y = \frac{\sin (x + y)}{\sin x \sin y} \][/tex]
To verify this, let's use the tangent and sine addition formulas:
[tex]\[ \tan x \tan y = \frac{\sin x \sin y}{\cos x \cos y} \][/tex]
[tex]\[ 1 - \tan x \tan y = 1 - \frac{\sin x \sin y}{\cos x \cos y} = \frac{\cos x \cos y - \sin x \sin y}{\cos x \cos y} \][/tex]
[tex]\[ \frac{\sin (x + y)}{\sin x \sin y} = \frac{\sin x \cos y + \cos x \sin y}{\sin x \sin y} \][/tex]
Since these expressions don't simplify to a common form between both sides, this is not an identity.
### Conclusion
From the above checks, the expressions that are identities are:
- Expression B: [tex]\(\tan x - \tan y = \frac{\sin (x - y)}{\cos x \cos y}\)[/tex]
- Expression C: [tex]\(\cos\left(x + \frac{\pi}{6}\right) = -\sin\left(x - \frac{\pi}{3}\right)\)[/tex]
Thus, the identities are [tex]\(\boxed{B \text{ and } C}\)[/tex].
