Determine which of the following are trigonometric identities.
I. $\frac { \cos ( \mathrm { t } ) - \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) - \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 0$
II. $\frac { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 1$
III. $\frac { \cos ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) \sin ( \mathrm { s } ) } = \cos ( \mathrm { s } ) + \sin ( \mathrm { t } )$
A) I is the only identity.
B) I, II, and III are identities.
C) II is the only identity.
D) II and II are the only identities.
E) I and II are the only identities.

Question

Quizplus · Accepted Answer

We can simplify identity I by using the identities $\sin(t) - \sin(s) = -\sin(s-t)$ and $\cos(t) -\cos(s) = - 2\sin(\frac{t+s}{2})\sin(\frac{t-s}{2})$:

$$\begin{aligned} \frac { \cos ( \mathrm { t } ) - \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) - \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } &= \frac{-2\sin(\frac{t+s}{2})\sin(\frac{t-s}{2})}{\sin(t)+\sin(s)} + \frac{-\sin(s-t)}{\cos(t)+\cos(s)} \ &= \frac{-2\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) (\cos(t)+\cos(s)) - \sin(s-t)(\sin(t)+\sin(s))}{(\sin(t)+\sin(s))(\cos(t)+\cos(s))} \end{aligned}$$
Next, we use the identity $\cos(A-B) = \cos A\cos B+\sin A\sin B$ to simplify the numerator:

$$\begin{aligned} -2&\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) (\cos(t)+\cos(s)) - \sin(s-t)(\sin(t)+\sin(s)) \ &= -2\sin(\frac{t+s}{2})\sin(\frac{t-s}{2})\cos t - 2\sin(\frac{t+s}{2})\sin(\frac{t-s}{2})\cos s - \sin(s-t)\sin t - \sin(s-t)\sin s \ &= -2\sin(\frac{t+s}{2})\sin(\frac{t-s}{2})\cos(\frac{t+s}{2})\cos(\frac{t-s}{2}) -2\sin(\frac{t+s}{2})\sin(\frac{t-s}{2})\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) - \sin(s-t)\sin t - \sin(s-t)\sin s \ &= 2\cos(\frac{t+s}{2})\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) - (\cos^2(\frac{t-s}{2})-\sin^2(\frac{t-s}{2}))\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) - \sin(s-t)\sin t - \sin(s-t)\sin s \ &= 2\cos(\frac{t+s}{2})\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) - \cos(\frac{t+s}{2})\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) - \cos(\frac{t-s}{2})\sin(\frac{t+s}{2})\sin(\frac{t-s}{2}) - \sin(s-t)\sin t - \sin(s-t)\sin s \ &= \cos(\frac{t+s}{2})\sin(\frac{t+s}{2})(2-\cos(\frac{t-s}{2})) - \sin(s-t)(\sin t+\sin s) \end{aligned}$$

Thus, we have:
$$\begin{aligned} \frac { \cos ( \mathrm { t } ) - \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) - \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } &= \frac{-\cos(\frac{t+s}{2})\sin(\frac{t+s}{2})(\cos(\frac{t-s}{2})-2) + \sin(s-t)(\sin(t)+\sin(s))}{(\sin(t)+\sin(s))(\cos(t)+\cos(s))} \end{aligned}$$

For this expression to be an identity, the numerator must be zero. We see that $\sin(s-t)(\sin(t)+\sin(s))$ cannot be zero for all values of $t$ and $s$ since $\sin$ is not identically zero. Therefore, we must have $\cos(\frac{t-s}{2})=2$. However, $\cos$ is bounded above by 1, so this is not possible. Therefore, the expression cannot be an identity, and the answer is A, I is the only identity.

Determine Which of the Following Are Trigonometric Identities II III