Which of the following expressions are not equal to 1?
A) sin^2 theta + cot^2 theta sin^2 theta
B) (sin^2 theta/1-cos theta)-cos theta
C) sec^2 theta + tan^2 theta
D) cot^2 theta sin^2 theta/cos^2theta
Answer: D
for D I get
c^2/s^2 * s^2 /c^2 = 1 sorry
A gives
s^2 (1 + c^2/s^2) = s^2 + c^2 = 1
B gives (does not make sense the way you wrote it)
Maybe you mean
sin^2 theta/(1-cos theta)-cos theta
[s^2 -c (1-c)] /(1-c)
[ s^2 + c^2 -c] / (1-c)
(1-c) / (1-c)
1
C gives
(1/c^2) + s^2/c^2
= (1 + s^2)/c^2
not 1
To determine which expressions are not equal to 1, we can simplify each expression and check if the simplified form is equal to 1.
Let's start by simplifying each expression:
A) sin^2(theta) + cot^2(theta) sin^2(theta)
- Rearranging terms, we get: sin^2(theta) * (1 + cot^2(theta))
- Since cot^2(theta) is equal to 1/tan^2(theta), we can substitute it in the expression:
= sin^2(theta) * (1 + 1/tan^2(theta))
B) (sin^2(theta)/1 - cos(theta)) - cos(theta)
- Simplifying the denominator, we have: 1 - cos(theta)
- Applying the formula for (a - b)(a + b), we can rewrite the expression: sin^2(theta)/ (1 - cos(theta)) - cos(theta)
= sin^2(theta)/ (1 - cos(theta)) - cos(theta)(1 - cos(theta))/(1 - cos(theta))
C) sec^2(theta) + tan^2(theta)
- Since sec^2(theta) is equal to 1/cos^2(theta) and tan^2(theta) is equal to sin^2(theta)/cos^2(theta), we can substitute them in the expression:
= (1/cos^2(theta)) + (sin^2(theta)/cos^2(theta))
D) cot^2(theta) sin^2(theta) / cos^2(theta)
- Since cot^2(theta) is equal to 1/tan^2(theta) and tan^2(theta) is equal to sin^2(theta)/cos^2(theta), we can substitute them in the expression:
= (1/tan^2(theta)) * sin^2(theta) / cos^2(theta)
Next, let's simplify each expression further:
A) sin^2(theta) * (1 + 1/tan^2(theta))
- Applying the formula for 1/tan^2(theta) = cos^2(theta)/sin^2(theta), we have:
= sin^2(theta) * (1 + cos^2(theta)/sin^2(theta))
= sin^2(theta) * (sin^2(theta) + cos^2(theta)) / sin^2(theta)
- Since sin^2(theta) + cos^2(theta) = 1, this simplifies to:
= sin^2(theta)
B) sin^2(theta) / (1 - cos(theta)) - cos(theta)(1 - cos(theta))/(1 - cos(theta))
- Simplifying the denominator, we have:
= sin^2(theta) / (1 - cos(theta)) - cos(theta)(1)/(1 - cos(theta))
= sin^2(theta) / (1 - cos(theta)) - cos(theta)/(1 - cos(theta))
- Finding a common denominator, we get:
= (sin^2(theta) - cos(theta))/(1 - cos(theta))
C) (1/cos^2(theta)) + (sin^2(theta)/cos^2(theta))
- Adding the fractions, we have:
= (1 + sin^2(theta))/cos^2(theta)
D) (1/tan^2(theta)) * sin^2(theta) / cos^2(theta)
- Multiplying the fractions, we get:
= sin^4(theta) / cos^4(theta)
Now, let's check if each simplified expression is equal to 1:
A) sin^2(theta) = 1
- Since sin^2(theta) can take any value between 0 and 1, this expression can be equal to 1.
B) (sin^2(theta) - cos(theta))/(1 - cos(theta)) =/= 1
- This expression is not always equal to 1. For example, when theta = 0, the expression becomes (0 - 1)/(1 - 1) = -1/0, which is undefined.
C) (1 + sin^2(theta))/cos^2(theta) = 1
- This expression simplifies to 1 by applying the Pythagorean identity sin^2(theta) + cos^2(theta) = 1.
D) sin^4(theta) / cos^4(theta) =/= 1
- This expression is not always equal to 1. For example, when theta = 90 degrees, the expression becomes (1/0) / (0/1) = undefined.
Therefore, the expression not equal to 1 is D) cot^2(theta) sin^2(theta) / cos^2(theta).