Simplify sin 𝜃 times cos 𝜃 times tan 𝜃 plus cot 𝜃.
So first, we just copy down the expression we need to simplify from the question. There are six trigonometric functions that we tend to use: sin, cos, tan, csc, sec, and cot. And they can all be written in terms of sin and cos alone. And it’s helpful to do that when we’re simplifying, so let’s use these relations to write our expression that we have to simplify in terms of sin and cos alone.
Well obviously, with sin and cos 𝜃, we don’t have anything to do, but we have to do something with tan 𝜃. We can rewrite tan 𝜃 as sin 𝜃 over cos 𝜃, like so.
We can do the same for cot 𝜃; cot 𝜃 is equal to cos 𝜃 over sin 𝜃, like so. And now we can see that our expression we need to simplify is only written in terms of sin and cos.
Here we have the sum of fractions, which we’d like to simplify by combining them into one fraction, and that requires finding a common denominator. We can see that cos 𝜃 sin 𝜃 will work for this. With this common denominator, the first fraction becomes sin 𝜃 times sin 𝜃 over cos 𝜃 times sin 𝜃.
And the second fraction becomes cos 𝜃 times cos 𝜃 over cos 𝜃 times sin 𝜃. And we can combine these two fractions now that they have a common denominator, cos 𝜃 sin 𝜃, to get sin 𝜃 sin 𝜃 plus cos 𝜃 cos 𝜃 over cos 𝜃 sin 𝜃.
We can do some cancellation here. We’re multiplying by cos 𝜃 and then dividing by it, and similarly we’re multiplying by sin 𝜃 and then dividing it. And so we’re left with sin 𝜃 sin 𝜃 plus cos 𝜃 cos 𝜃, which we can write as sin squared 𝜃 plus cos squared 𝜃.
And we recognize this line because it’s part of a famous identity: sin squared 𝜃 plus cos squared 𝜃 is equal to one. After all that simplification, we have a very simple answer: sin 𝜃 times cos 𝜃 times tan 𝜃 plus cot 𝜃 is equal to one.