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 establish the identity, let's start by writing the left side of the equation in terms of sine and cosine: \[ \frac{4 \sec \theta}{\csc \theta} + \frac{2 \sin \theta}{\cos \theta} \] We know that \(\sec \theta = \frac{1}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\). Substituting these into the equation, we get: \[ \frac{4}{\cos \theta} \cdot \frac{\sin \theta}{1} + \frac{2 \sin \theta}{\cos \theta} \] Now, we can rewrite the first fraction to have a common denominator with the second fraction: \[ \frac{4 \sin \theta}{\cos \theta} + \frac{2 \sin \theta}{\cos \theta} \] Now, we can add the fractions since they have a common denominator: \[ \frac{4 \sin \theta + 2 \sin \theta}{\cos \theta} \] This simplifies to: \[ \frac{6 \sin \theta}{\cos \theta} \] Now, we can rewrite this expression in terms of tangent, which is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\): \[ 6 \cdot \frac{\sin \theta}{\cos \theta} = 6 \tan \theta \] The expression from the previous step simplifies to \(6 \tan \theta\) using the **Quotient Identity**. The quotient identity states that for any angle \(\theta\): \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Therefore, the correct answer is: D. Addition and a Quotient Identity