Sum-to-product identity first, then separate the variables

1. A.
   $\log\tan\frac{y}{2} = C - 2\sin x$
2. B.
   $\log\tan\frac{y}{4} = C - 2\sin\frac{x}{2}$
3. C.
   $\log\tan\left(\frac{y}{2}+\frac{\pi}{4}\right) = C - 2\sin x$
4. D.
   $\log\tan\left(\frac{y}{2}+\frac{\pi}{4}\right) = C - 2\sin\frac{x}{2}$

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1. A.
   $\log tan\left( \frac{y}{2} \right) = c - 2\sin\frac{x}{2}$, where c is the constant of integration
2. B.
   $\log tan\left( \frac{y}{4} \right) = c - 2\sin\left( \frac{x}{2} \right)$, where $c$ is the constant of integration
3. C.
   $\ \log\left\lbrack \tan\left( \frac{y}{2} + \frac{\pi}{4} \right) \right\rbrack = c - 2\sin x$, where $c$ is the constant of integration
4. D.
   $\ \log\left\lbrack \tan\left( \frac{y}{4} + \frac{\pi}{4} \right) \right\rbrack = c - 2\sin\frac{x}{2}$, where $c$ is the constant of integration

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1. A.
   $\cos 2x + \cos 2y$
2. B.
   $\sec y + \tan y$
3. C.
   $-\dfrac{1}{3}\log\left|\tan\left(\dfrac{3y}{2}\right)\right|$
4. D.
   $2\log|\operatorname{\csc}3y - \cot 3y|$

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