Animated Solution for Mathematics - Trigonometry: If $\alpha +\beta =\pi /2$ and $\beta +\gamma =\alpha$, then $\tan \alpha$ equals

We are given two primary constraints for the angles $\alpha$, $\beta$, and $\gamma$: 1. $\alpha +\beta =\frac{\pi }{2}$ 2. $\beta +\gamma =\alpha$

These equations define the geometric relationship between the angles. Our goal is to express $\tan \alpha$ in terms of $\tan \beta$ and $\tan \gamma$.

Starting with the first condition, $\alpha +\beta =\frac{\pi }{2}$, we recognize these as complementary angles. Taking the tangent of both sides, we have:

Using the fundamental co-function identity $\tan \left(\frac{\pi }{2}-\theta \right)=\cot \theta$, we obtain $\tan \beta =\cot \alpha$. Since $\cot \alpha =\frac{1}{\tan \alpha }$, we arrive at the following identity:

Next, we address the second condition: $\beta +\gamma =\alpha$. Rearranging this gives $\gamma =\alpha -\beta$. Applying the tangent function to both sides, we use the compound angle formula:

We now substitute the identity $\tan \alpha \tan \beta =1$ into the denominator of our expression for $\tan \gamma$:

This simplifies to:

Multiplying both sides by $2$, we get $2\tan \gamma =\tan \alpha -\tan \beta$. Adding $\tan \beta$ to both sides, we reach the final result:

By trusting the trigonometric identities and following the logical flow, the relationship between the angles is revealed as a simple, elegant truth.