Multiple Choice
Which expression is NOT equivalent to 1? (theta) means 0 with dash in it.
A.)sin^2 (theta)+cot^2 (theta) sin^2 (theta)
B.)(sin^(theta))/(1-cos(theta))-cos(theta)
C,)sec^2 (theta)+ tan^2 (theta)
D.)(cot^2(theta) sin^2 (theta))/(cos^2(theta))
To find the expression that is NOT equivalent to 1, we need to simplify each expression and see if it simplifies to 1.
Let's evaluate each option:
A.) sin^2 (theta) + cot^2 (theta) sin^2 (theta)
Using the trigonometric identity, cot^2(theta) = 1 + tan^2(theta), we can rewrite the expression as:
sin^2 (theta) + (1 + tan^2(theta)) sin^2 (theta)
Factoring out sin^2(theta), we get:
sin^2(theta) + sin^2(theta) + tan^2(theta) sin^2(theta)
Combining the like terms, we have:
2sin^2(theta) + tan^2(theta) sin^2(theta)
This expression does not equal 1, so option A is not equivalent to 1.
B.) (sin^(theta))/(1-cos(theta))-cos(theta)
To simplify this expression, we need to rationalize the denominator:
(sin^(theta))/(1-cos(theta))-cos(theta) * ((1+cos(theta))/(1+cos(theta)))
Expanding, we get:
(sin^2(theta) + sin^2(theta)cos(theta))/(1 - cos^2(theta)) - cos(theta)
Using the identity 1 - cos^2(theta) = sin^2(theta), we can rewrite the expression as:
(sin^2(theta) + sin^2(theta)cos(theta))/sin^2(theta) - cos(theta)
Canceling out sin^2(theta), we have:
1 + cos(theta)
This expression does equal 1, so option B is not the correct answer.
C.) sec^2 (theta) + tan^2 (theta)
Since sec^2(theta) = 1 + tan^2(theta), we can rewrite the expression as:
1 + tan^2(theta) + tan^2(theta)
Simplifying, we get:
1 + 2tan^2(theta)
This expression does not equal 1, so option C is not equivalent to 1.
D.) (cot^2(theta) sin^2(theta))/(cos^2(theta))
Using the identity cot(theta) = 1/tan(theta), we can rewrite the expression as:
(1/tan^2(theta)) * sin^2(theta) / cos^2(theta)
Simplifying, we have:
sin^2(theta) / (tan^2(theta) * cos^2(theta))
Using the trigonometric identity tan^2(theta) = sin^2(theta) / cos^2(theta), we can rewrite the expression as:
(sin^2(theta)) / (sin^2(theta))
Canceling out sin^2(theta), we have:
1
This expression equals 1, so option D is the correct answer.
Therefore, the expression NOT equivalent to 1 is (D) (cot^2(theta) sin^2(theta))/(cos^2(theta)).