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Intensive Precalculus Homework

 
 
Picture of Jyoti Mujumdar
Intensive Precalculus Homework
by Jyoti Mujumdar - Saturday, June 13, 2020, 3:28 PM
 

After watching the video lecture for section 5.4, I didn't understand how to verify a cofunction identity. Can someone please help me with this? Thanks!

 
Picture of Dr. Kevin Wang
Re: Intensive Precalculus Homework
by Dr. Kevin Wang - Saturday, June 13, 2020, 11:27 PM
 
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.