Simplify tan minus 𝜃 times csc 𝜃.
We use the fact that tan is an odd function to write tan minus 𝜃 as minus tan
𝜃. We now write everything in terms of sin 𝜃 and cos 𝜃. Tan 𝜃 is sin 𝜃 over cos 𝜃 and csc 𝜃 is one over sin 𝜃. So the expression we need to simplify becomes minus sin 𝜃 over cos 𝜃 times one over
sin 𝜃. We can see that the sin 𝜃 in the numerator cancels with the sin 𝜃 in the
denominator, and we’re left with just minus one over cos 𝜃. And we know that sec 𝜃 is one over cos 𝜃, so we can simplify further to get our
final answer minus sec 𝜃.
To recap, we use the fact that tan is an odd function. And we can see that tan is an odd function by considering its graph, which has
180-degree rotational symmetry about the origin. Alternatively, we can prove it using this identity here and the fact that sin is an
odd function and cos is an even function.
Using this identity, we see that tan minus 𝜃 is equal to sin minus 𝜃 over cos minus 𝜃. Sin is an odd function, so sin minus 𝜃 is equal to minus sin 𝜃. And cos is an even function, so cos minus 𝜃 is equal to cos 𝜃. And as sin 𝜃 over cos 𝜃 is tan 𝜃, minus sin 𝜃 over cos 𝜃 is minus tan 𝜃, and
tan minus 𝜃 is equal to minus tan 𝜃.
This is the fact we used for our first step. After this first step, we rewrote tan 𝜃 and csc 𝜃 in terms of sin 𝜃 and cos
𝜃. And after some cancellation, we use another related identity that sec 𝜃 is equal to
one over cos 𝜃. In fact, the only identity of this type that we didn’t need was cot 𝜃 equals cos 𝜃
over sin 𝜃.