Even and Odd in a Proof

When a negative angle sits inside a proof, clear it first: the rest of the work then runs in angle x alone. Take $\cos\left(-x\right)\tan\left(x\right)$. Cosine is even, so it becomes $\cos\left(x\right)\tan\left(x\right)$. Write tan as a quotient, $\cos\left(x\right)\cdot \frac{\sin\left(x\right)}{\cos\left(x\right)}$, multiply the tops and the bottoms, and cancel cos x. What is left is sin x.

With an odd function the minus comes out in front and rides along to the end: $\sin\left(-x\right)\csc\left(x\right)$ is $-\sin\left(x\right)\csc\left(x\right)$, which is $-1$. Drop that minus and the last line is wrong.