prove that tanA divided by the sinA equals the secA

Question

prove that tanA divided by the sinA equals the secA

Answer 1

tanA = sinA/cosA

so tanA / sinA
= (sinA/cosA)/sinA
= 1/cosA
= secA

Answer 2

(sinA/cosA)/ sinA

= 1/cosA
= secA

Answer 3

To prove that tan(A) / sin(A) = sec(A), we need to use the basic trigonometric identities.

Let's start with the definition of tangent (tan):
tan(A) = sin(A) / cos(A)

Now, let's substitute this value for tan(A) into the expression tan(A) / sin(A):
tan(A) / sin(A) = (sin(A) / cos(A)) / sin(A)

Next, we can simplify the expression by multiplying both sides of the fraction by the reciprocal of sin(A), which is 1/sin(A):
tan(A) / sin(A) = (sin(A) / cos(A)) * (1 / sin(A))

Now, we can cancel out the sin(A) terms:
tan(A) / sin(A) = (1 / cos(A))

Finally, recall the definition of secant (sec):
sec(A) = 1 / cos(A)

So, we can see that tan(A) / sin(A) is indeed equal to sec(A).