Factorthen use fundamental identities to simplify the expression below and determine which of the following is not equivalent cot^2 a * tan^2 a + cot^2 a A. csc^ 2 alpha B.1/ sin^ 2 alpha C.1/ 1-cos^ 2 alpha D.sec^ 2 alpha E.1+cot^ 2 alpha
cot^2 a * tan^2 a + cot^2 a
cot²a(tan²a+1)
Recall that
Tan²a+1=1/cos²a
1/sin²a=cosec²a
So
(Cos²a/sin²a)/cos²a
1/sin²a=cosec²a
To simplify the expression cot^2(a) * tan^2(a) + cot^2(a), we need to use the fundamental identities for cotangent (cot) and tangent (tan).
The fundamental identities are:
1. cot^2(a) = 1 + tan^2(a)
2. 1 + cot^2(a) = csc^2(a)
3. sec^2(a) = 1 + tan^2(a)
4. 1 + tan^2(a) = sec^2(a)
5. 1 - cos^2(a) = sin^2(a)
Let's start simplifying the expression:
cot^2(a) * tan^2(a) + cot^2(a)
Using the fundamental identity 1 (cot^2(a) = 1 + tan^2(a)), we can substitute cot^2(a) as (1 + tan^2(a)):
(1 + tan^2(a)) * tan^2(a) + (1 + tan^2(a))
Simplifying further, distribute the terms:
tan^2(a) + tan^4(a) + 1 + tan^2(a)
Combine like terms:
tan^4(a) + 2tan^2(a) + 1
Now, based on the answer choices, let's determine which one is not equivalent to the expression:
A. csc^2(alpha)
B. 1/sin^2(alpha)
C. 1/(1 - cos^2(alpha))
D. sec^2(alpha)
E. 1 + cot^2(alpha)
To compare, let's convert the expression to alpha instead of a:
tan^4(alpha) + 2tan^2(alpha) + 1
Looking at the options:
A. csc^2(alpha) = 1/sin^2(alpha) - This is equivalent because sin^2(alpha) = tan^2(alpha) + 1.
B. 1/sin^2(alpha) - This is equivalent because sin^2(alpha) = tan^2(alpha) + 1.
C. 1/(1 - cos^2(alpha)) = sec^2(alpha) - This is equivalent because sec^2(alpha) = tan^2(alpha) + 1.
D. sec^2(alpha) - This is equivalent because sec^2(alpha) = tan^2(alpha) + 1.
E. 1 + cot^2(alpha) - This is different from the expression we simplified.
Therefore, the answer choice that is not equivalent to the expression is option E. 1 + cot^2(alpha).
To simplify the expression cot^2(a) * tan^2(a) + cot^2(a), we can start by using the fundamental identity:
1 + cot^2(a) = csc^2(a)
By substituting cot^2(a) into the above identity, we get:
1 + cot^2(a) = 1 + csc^2(a)
Now, we can substitute this back into the original expression:
cot^2(a) * tan^2(a) + cot^2(a) = cot^2(a) * tan^2(a) + (1 + csc^2(a))
Next, we notice that cot^2(a) * tan^2(a) is equivalent to 1, so we can substitute that into the expression:
1 + (1 + csc^2(a)) = 2 + csc^2(a)
Therefore, the simplified expression is 2 + csc^2(a).
Now, we can determine which of the given options is not equivalent to this expression:
A. csc^2(alpha) - This is equivalent to 2 + csc^2(a). So, this option is equivalent.
B. 1/sin^2(alpha) - This is not equivalent to 2 + csc^2(a), as it does not have the "+2" term. So, this option is not equivalent.
C. 1/(1 - cos^2(alpha)) - This is equivalent to 2 + csc^2(a), as it simplifies to the same expression. So, this option is equivalent.
D. sec^2(alpha) - This is not equivalent to 2 + csc^2(a), as it does not have the "+2" term. So, this option is not equivalent.
E. 1 + cot^2(alpha) - This is equivalent to 2 + csc^2(a), as it simplifies to the same expression. So, this option is equivalent.
Therefore, the answer is B. 1/sin^2(alpha) since it is not equivalent to the simplified expression 2 + csc^2(a).