Find the value of cot squared 𝜃 given csc squared 𝜃 is equal to 25 over nine.
Now what we’re absolutely not going to do is solve the equation csc squared 𝜃 equals 25 over nine for 𝜃. Instead, we’re going to recall some trigonometric identities. The first identity we recall is that sin squared 𝜃 plus cos squared 𝜃 is equal to one. And we’re going to derive the next identity, though it is absolutely fine to simply quote it. We’re going to divide each term here by sin squared 𝜃. sin squared 𝜃 divided by sin squared 𝜃 is equal to one. Then, we recall that cot 𝜃 is equal to one over tan 𝜃. And tan 𝜃 is sin 𝜃 over cos 𝜃. So its reciprocal cot 𝜃 must be equal to cos 𝜃 over sin 𝜃.
This means that cos squared 𝜃 over sin squared 𝜃 must be equal to cot squared 𝜃. And then, since we know that csc 𝜃 is equal to one over sin 𝜃, we can say that this is equal to csc squared 𝜃. Now, it’s absolutely fine to quote this identity one plus cot squared 𝜃 equals csc squared 𝜃. To find an identity for cot squared 𝜃, I’m going to subtract one from both sides. And we find that cot squared 𝜃 is always equal to csc squared 𝜃 minus one.
Now, in our question, we’re told that csc squared 𝜃 is 25 over nine. So we can say that cot squared 𝜃 here must be equal to 25 over nine minus one. To evaluate this, let’s write one as nine over nine. And when we subtract nine over nine from 25 over nine, we get 16 over nine.
And we find that when csc squared 𝜃 is equal to 25 over nine, cot squared 𝜃 is 16 over nine.