Use the fundamental identities to simplify the expression:
cot beta sec beta
I used 1+tan^2u=secu since cot is the inverse of tan. I flipped the tangent, then so it was 1+ (1/tan). But the book's answer is the cosecant of beta. Where did this come from??
Fundamental identities include:
cot(β)=cos(β)/sin(β)
sec(β)=1/cos(β), and
csc(β)=1/sin(β)
I will let you take it from here.
Is the right answer 1/sinB?
That's correct, although 1/sin(β) is generally written as csc(β), as your answer indicates.
To simplify the expression cot β sec β, we can start by using the definitions of cotangent (cot β) and secant (sec β):
cot β = 1/tan β
sec β = 1/cos β
Substituting these definitions into the given expression, we have:
cot β sec β = (1/tan β) * (1/cos β)
Now, let's manipulate this expression using the fundamental trigonometric identity:
1 + tan² β = sec² β
We can rearrange this identity to get:
sec² β - tan² β = 1
Now, we can substitute the expression sec² β - tan² β with 1 in the original expression:
cot β sec β = (1/tan β) * (1/cos β)
= (1/tan β) * (1/cos β) * (sec² β - tan² β)
= (1/tan β) * (1/cos β) * 1 (since sec² β - tan² β = 1)
= 1/tan β
= csc β
Therefore, cot β sec β simplifies to csc β, which is the cosecant of β.