Establish the identity. \( \frac{4 \sec \theta}{\csc \theta}+\frac{2 \sin \theta}{\cos \theta}=6 \tan \theta \) Write the left side of the identity in terms of sine and cosine. Rewrite the numerator and denominator separately. Simplify the fraction from the previous step such that both the fractions have the common denominator cos \( \theta \). \( +\frac{2 \sin \theta}{\cos \theta} \) (Do not simplify.) The expression from the previous step then simplifies to 6 tan \( \theta \) using what? A. Addition and the Cancellation Property B. Addition and a Reciprocal Identity C. Addition and an Even-Odd Identity D. Addition and a Quotient Identity E. Addition and a Pythagorean Identity

To dive into the fun of identities, let's first transform the left side! The expression can be rewritten using sine and cosine as: \[ \frac{4 \sec \theta}{\csc \theta} = \frac{4 \cdot \frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} = \frac{4 \sin \theta}{\cos \theta} \] Now we have: \[ \frac{4 \sin \theta}{\cos \theta} + \frac{2 \sin \theta}{\cos \theta} = \frac{(4 + 2) \sin \theta}{\cos \theta} = \frac{6 \sin \theta}{\cos \theta} \] This gives us \(6 \tan \theta\) since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and we can express this sum as a single fraction! To show that it simplifies to \(6 \tan \theta\), we relied on the relationship between sine and cosine, which points us confidently toward option D - Addition and a Quotient Identity. This fulfills our identity quest with flair!