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

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a trigonometric identity; we are uncovering a hidden symmetry.

When you see a condition like $\alpha +\beta +\gamma =2\pi$, your brain might immediately jump to complex expansions. But wait—take a breath. Look at the options.

They are all in terms of half-angles. This is the universe whispering the secret to you: the problem is already half-solved if you just listen to the structure.

The Half-Angle Transformation

We start with the given condition: $\alpha +\beta +\gamma =2\pi$. Since the target expressions involve $\frac{\alpha }{2}$, $\frac{\beta }{2}$, and $\frac{\gamma }{2}$, we must transform our world.

Dividing the entire equation by $2$ gives us:

This is our new foundation. It is elegant, simple, and perfectly aligned with the trigonometric identities we are about to deploy.

The Strategic Shift

Now, we need to bring the tangent function into play. But we cannot just take the tangent of all three terms at once. We need a strategy.

Let us isolate two terms on the left and push the third to the right:

Why do this? Because we know exactly how to handle $\tan (\pi -\theta )$. It is a bridge to simplicity.

The Tangent Identity

Taking the tangent of both sides, we get $\tan (\frac{\alpha }{2}+\frac{\beta }{2})=\tan (\pi -\frac{\gamma }{2})$. On the left, we use the addition identity:

Substituting $A=\frac{\alpha }{2}$ and $B=\frac{\beta }{2}$, the left side becomes:

On the right, the identity $\tan (\pi -\theta )=-\tan \theta$ transforms $\tan (\pi -\frac{\gamma }{2})$ into $-\tan \frac{\gamma }{2}$. We have successfully reduced the complexity.

The Algebraic Dance

Now, we equate them:

To clear the fraction, we cross-multiply. Be careful here—this is where the battle is often won or lost. We get:

Expanding the right side and distributing the negative sign, we obtain:

Finally, moving $-\tan \frac{\gamma }{2}$ to the left, we arrive at the beautiful conclusion:

This is the elegance of mathematics. We started with a simple sum and ended with a profound relationship between the sum and the product of tangents. You have conquered this problem!