IDNLearn.com connects you with a community of knowledgeable individuals ready to help. Ask your questions and receive comprehensive and trustworthy answers from our experienced community of professionals.
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].
### 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].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.