Which of the following expressions is equivalent to one over sec 𝜃? The options are sin 𝜃, cos 𝜃, tan 𝜃, cot 𝜃, or cosec 𝜃.
We will begin by recalling what is meant by sec of 𝜃. Sec of 𝜃 is one of our reciprocal trigonometric functions. It’s defined as the reciprocal of cos 𝜃. That’s one over cos 𝜃. So the expression we’ve been given in the question, which is one over sec of 𝜃, means one divided by one over cos 𝜃.
We recall that, to divide by a fraction, we flip or invert that fraction. And then, we multiply. So one divided by one over cos 𝜃 is equal to one multiplied by cos 𝜃 over one. This just simplifies to cos of 𝜃. So this is our answer for which expression is equivalent to one over sec 𝜃.
In fact, we could’ve worked this out without doing the division because reciprocal relationships work both ways. If sec 𝜃 is the reciprocal of cos 𝜃, then cos 𝜃 is also the reciprocal of sec 𝜃. So one over sec 𝜃 is equal to cos 𝜃.
Let’s just recall what is meant by two of the other options we were given, cot of 𝜃 and cosec of 𝜃, which are the reciprocal trigonometric functions of tan 𝜃 and sin 𝜃, respectively. To help you remember which reciprocal trigonometric function belongs with which trigonometric function, we could look at the third letters. For example, the third letter of cot is t. And cot of 𝜃 is equal to one over tan 𝜃. The first letter of tan is also t. This same method works for cosec of 𝜃, which is equal to one over sin 𝜃, and sec of 𝜃, which is equal to one over cos of 𝜃.
Our answer to which expression is equivalent to one over sec 𝜃 is cos 𝜃.