Which of the following are identities? Check all that apply.
(Points : 2)
sin2x = 1 - cos2x
sin2x - cos2x = 1
tan2x = 1 + sec2x
cot2x = csc2x - 1
Question 4. 4. Which of the following equations are identities? Check all that apply.
(Points : 2)
Question 5. 5. The expression sinx(cscx - cotx cosx) can be simplified to _____.
(Points : 2)
sin2x
sin2x - cscx
cos2x
sinx - tanx
Question 6. 6. The expression (secx + tanx)2 is the same as _____.
(Points : 2)
sec2x + tan2x
sec2x + 2cscx + tan2x
1 + 2cscx
1 + 2tan2x + 2secx tanx
Question 7. 7. Which of the following would be an acceptable first step in simplifying the expression ?
(Points : 2)
tanx + sinx
Question 8. 8. All identities are equations, and all equations are identities.
(Points : 2)
True
False
Question 9. 9. Which of the following is an identity?
(Points : 2)
sin2x - cos2x = 1
csc2x + cot2x = 1
(cscx + cotx)2 = 1
sin2x sec2x + 1 = cot2x csc2x
Question 10. 10. Which of the following is not an identity?
(Points : 2)
cos2x cscx - cscx = -sinx
sinx(cotx + tanx) = secx
cos2x - sin2x = 1- 2sin2x
csc2x + sec2x = 1
Hey, there is a limit !
I will do the first one, then you try the rest.
Which of the following are identities? Check all that apply.
(Points : 2)
sin2x = 1 - cos2x
I assume you mean sin^2 x not sin 2x
This is the same as
sin^2 x + cos^2 x = 1
which IS an identity.
sin^2x - cos^2x = 1 No way
tan2x = 1 + sec2x
sin^2 x/cos^2x = 1 + 1/cos^2 x
is
sin^2 x cos^2 x = 1 which is the same No way
cot2x = csc2x - 1
is
cos^2 x/sin^2 x = 1/sin^2 x -1
cos^2 x = 1 - sin^2 x
cos^2 x + sin^2 x = 1 YES
Here are the answers to the questions:
Question 1: The following is an identity: sin2x = 1 - cos2x
Question 2: None of the equations listed are identities
Question 3: The simplified expression of sinx(cscx - cotx cosx) is sin2x - cscx
Question 4: The expression (secx + tanx)2 is the same as sec2x + 2tan2x + 2secx tanx
Question 5: An acceptable first step in simplifying the expression is tanx + sinx
Question 6: False. Not all equations are identities and vice versa.
Question 7: The following is an identity: sin2x - cos2x = 1
Question 8: The following is not an identity: cos2x cscx - cscx = -sinx
To determine which of the given expressions are identities, we need to check if the expressions hold true for all possible values of the variable (x) in the given domain.
Let's go through each question one by one.
Question 1:
The expression sin2x = 1 - cos2x is an identity. This is because it holds true for all values of x.
Question 2:
The expression sin2x - cos2x = 1 is an identity. This is because it holds true for all values of x.
Question 3:
The expression tan2x = 1 + sec2x is not an identity. This is because it does not hold true for all values of x.
Question 4:
The given options are missing, so we cannot provide an answer for this question.
Question 5:
To simplify the expression sinx(cscx - cotx cosx), we distribute the sinx term:
sinx * cscx - sinx * cotx * cosx
This simplifies to:
1 - cos2x
Therefore, the simplified expression is sin2x.
Question 6:
To expand the expression (secx + tanx)2, we use the formula for squaring a binomial:
(secx + tanx)2 = (secx)2 + 2(secx)(tanx) + (tanx)2
This simplifies to:
sec2x + 2secxtanx + tan2x
Therefore, the expanded expression is sec2x + 2secxtanx + tan2x.
Question 7:
The question is incomplete, so we cannot provide an answer for this question.
Question 8:
The statement "All identities are equations, and all equations are identities" is False. While all identities can be considered equations, not all equations are identities. Identities are equations that hold true for all values of the variables.
Question 9:
The expression sin2x - cos2x = 1 is an identity. This is because it holds true for all values of x.
Question 10:
The expression cos2x - sin2x = 1 - 2sin2x is not an identity. This is because it does not hold true for all values of x.
The expressions csc2x + cot2x = 1 and sin2x sec2x + 1 = cot2x csc2x are identities. These hold true for all values of x.