1. The problem is to solve the equation involving the tangent function: $\tan\left(\frac{\pi}{2} x\right)$ and $2 \tan\left(t - \frac{\pi}{2}\right)$. We need to understand what is being asked, but since the user asks "How do I solve what the equation?", we assume the goal is to solve for $x$ or $t$ in an equation involving these expressions. 2. Recall the tangent function properties and identities: - $\tan\left(\theta - \frac{\pi}{2}\right) = -\cot(\theta)$. - The tangent function has vertical asymptotes at $\theta = \frac{\pi}{2} + k\pi$, $k \in \mathbb{Z}$. 3. Using the identity for the second term: $$ 2 \tan\left(t - \frac{\pi}{2}\right) = 2 \cdot (-\cot t) = -2 \cot t $$ 4. If the equation is $\tan\left(\frac{\pi}{2} x\right) = 2 \tan\left(t - \frac{\pi}{2}\right)$, then substituting gives: $$ \tan\left(\frac{\pi}{2} x\right) = -2 \cot t $$ 5. Express $\cot t$ as $\frac{1}{\tan t}$: $$ \tan\left(\frac{\pi}{2} x\right) = -\frac{2}{\tan t} $$ 6. Multiply both sides by $\tan t$: $$ \tan t \cdot \tan\left(\frac{\pi}{2} x\right) = -2 $$ 7. This is the simplified relation between $x$ and $t$. To solve for one variable, you need additional information or constraints. Summary: - Use the identity $\tan\left(t - \frac{\pi}{2}\right) = -\cot t$. - Rewrite the equation accordingly. - Multiply to clear denominators and simplify. Without more context or a specific equation to solve, this is the general approach to handle the given expressions.