what is the correct answer in proving trigonometric identities if tan theta/sin theta = sec theta
To prove the trigonometric identity
tan(θ) / sin(θ) = sec(θ),
we need to manipulate the left side of the equation until it is equivalent to the right side using known trigonometric identities.
Let's start by rewriting tan(θ) as sin(θ) / cos(θ) and the reciprocal of sec(θ), which is cos(θ):
(sin(θ) / cos(θ)) / sin(θ) = cos(θ).
Now, we can simplify the left side of the equation. When dividing by a fraction, we can multiply by its reciprocal. In this case, we multiply by the reciprocal of sin(θ), which is 1/sin(θ):
(sin(θ) / cos(θ)) * (1 / sin(θ)) = cos(θ).
Simplifying the expression:
(sin(θ) * (1 / sin(θ))) / cos(θ) = cos(θ).
The sin(θ) in the numerator cancels out with the denominator:
1 / cos(θ) = cos(θ).
Finally, we use the reciprocal identity of cosine, which states that 1/cos(θ) is equal to sec(θ):
sec(θ) = cos(θ).
Therefore, we have proven that tan(θ) / sin(θ) is equal to sec(θ).