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Intensive Precalculus Homework
Re: Intensive Precalculus Homework
In general, verifying an identity involves using the sum or difference formulas to simplify. ("Cofunctions" mean that "sin" and "cos", "tan" and "cot", "sec" and "csc"--in each pair, one is the cofunction of the other.) For example, to verify $\sin\left(\dfrac{\pi}{2}-x\right)=\cos x$, we could do
$$\sin\left(\dfrac{\pi}{2}-x\right) = \sin\dfrac{\pi}{2}\cos x - \cos\dfrac{\pi}{2}\sin x = 1\cdot\cos x - 0\cdot\sin x=\cos x.$$
In the homework, we are required to verify that $\tan\left(\dfrac{\pi}{2}-x\right)=\cot x$. This is a problem because if we tried to do the above, we encounter $\tan\dfrac{\pi}{2}$ which is undefined. To avoid this, you can write
$$\tan\left(\dfrac{\pi}{2}-x\right) = \frac{\sin\left(\dfrac{\pi}{2}-x\right)}{\cos\left(\dfrac{\pi}{2}-x\right)}$$
and go from there.
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