\( \cos ^ { 2 } \alpha + ( \frac { \cos \alpha } { \tan \alpha } ) ^ { 2 } = \frac { 1 } { \tan ^ { 2 } a } \)

Let’s break down the equation and see how it can be simplified! First, we know that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and therefore \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \). This means that \( \frac{1}{\tan^2 \alpha} = \frac{\cos^2 \alpha}{\sin^2 \alpha} \). Now, let’s simplify the left side step by step. First, we take \( \frac{\cos \alpha}{\tan \alpha} \) which simplifies to \( \frac{\cos^2 \alpha}{\sin \alpha} \). So squaring that gives us \( \left( \frac{\cos^2 \alpha}{\sin \alpha} \right)^2 = \frac{\cos^4 \alpha}{\sin^2 \alpha} \). Combine this with the \( \cos^2 \alpha \) term: \[ \cos^2 \alpha + \frac{\cos^4 \alpha}{\sin^2 \alpha} \] Replace \( \sin^2 \alpha \) with \( 1 - \cos^2 \alpha \) since \( \sin^2 \alpha + \cos^2 \alpha = 1 \), leading to: \[ \cos^2 \alpha \cdot \frac{(1 - \cos^2 \alpha) + \cos^2 \alpha}{(1 - \cos^2 \alpha)} \] This can be quite complex—leading us toward further algebraic manipulation or trigonometric identities to reach simplification or solutions. Keep tinkering with those identities, and you might find the elegant solution hiding in there!