Use the fundamental identities to simplify the expression.
A)cot θ
B)cot θ
C)sec θ
D)cos θ
E)csc θ

Question

Use the fundamental identities to simplify the expression. ​  ​
A)cot θ 
B)​cot θ 
C)​sec θ 
D)​cos θ 
E)​csc θ

Quizplus · Accepted Answer

Using the fundamental identity for tangent, we have:
tan²θ + 1 = sec²θ
Subtracting 1 from both sides and dividing by tan²θ, we get:
sec²θ/tan²θ = 1/tan²θ
Recall that cotangent is the reciprocal of tangent, so we can write this as:
sec²θcot²θ = 1
Using the Pythagorean identity for sine and cosine, we have:
1 + cot²θ = csc²θ
Subtracting cot²θ from both sides and taking the reciprocal, we get:
csc²θ/cot²θ = 1/cot²θ
Multiplying both sides by cot²θ, we get:
csc²θ = cot²θ
Taking the square root of both sides, we get:
cscθ = |cotθ|
Recall that cosine is the reciprocal of sine, so we can write this as:
cosθ = |sinθ|/|cosθ|
Since absolute values do not affect the sign, this simplifies to:
cosθ = sinθ/cosθ
which is equivalent to:
cos²θ = sinθ
Therefore, the best choice is D) cos θ.

Use the Fundamental Identities to Simplify the Expression