Evaluate the following expression. $\frac { 3 + 3 \sin 3 \theta } { \cos 3 \theta } + \frac { 3 \cos 3 \theta } { 1 + \sin 3 \theta }$
A) $6 \sec 3 \theta$
B) $- 6 \sec 3 \theta$
C) $\sec 6 \theta$
D) $3 \sec 3 \theta$
E) $- 3 \sec 3 \theta$

Question

Quizplus · Accepted Answer

The given expression can be simplified as follows:$$\frac { 3 + 3 \sin 3 \theta } { \cos 3 \theta } + \frac { 3 \cos 3 \theta } { 1 + \sin 3 \theta } = 3 \left( \frac { 1 + \sin 3 \theta } { \cos 3 \theta } \right) + 3 \left( \frac { \cos 3 \theta } { 1 + \sin 3 \theta } \right)$$Notice that the second term is the reciprocal of the first term, and when multiplied, they give 1. However, since they are being added, we simplify the first term:$$3 \left( \frac { 1 + \sin 3 \theta } { \cos 3 \theta } \right) = 3 \left( \frac { \cos 3 \theta + \sin 3 \theta \cos 3 \theta } { \cos^2 3 \theta } \right)$$Using the identity $\sec \theta = \frac{1}{\cos \theta}$, the expression simplifies to:$$3 \sec 3 \theta + 3 \tan 3 \theta \sec 3 \theta = 3 \sec 3 \theta (1 + \tan 3 \theta)$$Given that $\tan 3 \theta$ and $1$ are not directly related in a way that simplifies to a single trigonometric function, the initial simplification approach was incorrect. The correct approach is recognizing that both terms in the original expression are structured to directly apply the secant function:$$\frac { 3 + 3 \sin 3 \theta } { \cos 3 \theta } + \frac { 3 \cos 3 \theta } { 1 + \sin 3 \theta } = 3\sec 3 \theta + 3\sec 3 \theta = 6\sec 3 \theta$$Thus, the correct answer is $6 \sec 3 \theta$, which corresponds to choice A.

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Evaluate the Following Expression $\frac { 3 + 3 \sin 3 \theta } { \cos 3 \theta } + \frac { 3 \cos 3 \theta } { 1 + \sin 3 \theta }$