Find the value of sec squared 𝜃 given tan squared 𝜃 equals four.
To answer this question, we’re not going to solve the equation tan squared 𝜃 equals four for 𝜃. Instead, we’re going to find a Pythagorean identity that links sec squared 𝜃 and tan squared 𝜃.
To find this identity, we’re going to recall one of the most straightforward trigonometric identities. That is, sin squared 𝜃 plus cos squared 𝜃 equals one. We also know that tan 𝜃 is equal to sin 𝜃 over cos 𝜃. So it follows that tan squared 𝜃 must also be equal to sin squared 𝜃 over cos squared 𝜃.
And so what we’re going to do is individually divide each of the elements in our first identity by cos squared 𝜃. When we do, we can write sin squared 𝜃 divided by cos squared 𝜃 is tan squared 𝜃. Cos squared 𝜃 divided by cos squared 𝜃 is one. And since one over cos 𝜃 is sec 𝜃, we know one over cos squared 𝜃 is sec squared 𝜃.
And so this is the Pythagorean identity we’re going to use. And in fact, we don’t need to derive it each time. We can simply quote that sec squared 𝜃 is equal to tan squared 𝜃 plus one. In this question, tan squared 𝜃 is equal to four. So we can say that sec squared 𝜃 must be equal to four plus one, which is simply five. And so the value of sec squared 𝜃 given that tan squared 𝜃 is equal to four is five.