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Which of the following are identities? Check all that apply.

A. [tex]\tan \left(x-\frac{\pi}{4}\right)=\tan x-1[/tex]

B. [tex]\tan x-\tan y=\frac{\sin (x-y)}{\cos x \cos y}[/tex]

C. [tex]\cos \left(x+\frac{\pi}{6}\right)=-\sin \left(x-\frac{\pi}{3}\right)[/tex]

D. [tex]1-\tan x \tan y=\frac{\sin (x+y)}{\sin x \sin y}[/tex]


Sagot :

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].