Animated Solution for Mathematics - Trigonometry: If ${\sin }^4\alpha +4{\cos }^4\beta +2=4\sqrt{2}\sin \alpha \cos \beta$; $\alpha ,\beta \in [0,\pi ]$, then $\cos (\alpha +\beta )-\cos (\alpha -\beta )$ is equal to :

Imagine you are standing before a complex trigonometric equation:

At first glance, it looks like a chaotic mess of powers and mixed variables. But in the world of JEE Advanced, chaos is often just order in disguise.

Today, we are going to peel back the layers of this problem using one of the most elegant tools in your mathematical arsenal: the Arithmetic Mean-Geometric Mean (AM-GM) Inequality.

Recognizing the Pattern

Whenever you see a sum of terms on one side and a product on the other, your intuition should immediately scream 'AM-GM!'

The inequality states that for any non-negative real numbers $x_1,x_2,...,x_n$, the arithmetic mean is greater than or equal to the geometric mean:

Our equation has ${\sin }^4\alpha$ and $4{\cos }^4\beta$. We have a constant $2$ sitting there, looking lonely. To make this work, we need four terms to match the fourth-degree nature of our variables.

Let us split that $2$ into $1+1$. Now, our left-hand side (LHS) is ${\sin }^4\alpha +4{\cos }^4\beta +1+1$.

Calculating the Means

Let us calculate the Arithmetic Mean (AM) of these four terms:

Now, let us look at the Geometric Mean (GM) of the same four terms:

Simplifying this is where the magic happens. The fourth root of $4$ is $\sqrt{2}$, and the fourth root of ${\sin }^4\alpha$ is simply $\sin \alpha$ (since $\alpha \in [0,\pi ]$, $\sin \alpha$ is non-negative).

Similarly, the fourth root of ${\cos }^4\beta$ is $|\cos \beta |$. Thus, $GM=\sqrt{2}\sin \alpha |\cos \beta |$.

The Equality Condition

By the AM-GM inequality, we know that $AM\ge GM$. Multiplying by $4$, we get:

Our given equation is an equality. In the realm of inequalities, equality is a rare and special state. It only occurs when all the terms involved are identical!

This forces our hand:

This is the breakthrough! From ${\sin }^4\alpha =1$, we find $\sin \alpha =1$, which means $\alpha =\frac{\pi }{2}$.

From $4{\cos }^4\beta =1$, we find ${\cos }^4\beta =\frac{1}{4}$, which means ${\cos }^2\beta =\frac{1}{2}$. Since $\beta \in [0,\pi ]$, $\sin \beta$ must be positive, so ${\sin }^2\beta =1-{\cos }^2\beta =\frac{1}{2}$, giving us $\sin \beta =\frac{1}{\sqrt{2}}$.

We are asked to find $\cos (\alpha +\beta )-\cos (\alpha -\beta )$. Using the sum-to-product identities, we know this expression simplifies beautifully to $-2\sin \alpha \sin \beta$.

Substituting our hard-won values:

And there it is. The complexity collapses into a simple, elegant constant. You didn't just solve an equation; you navigated a logical structure.

Keep this mindset—look for the symmetry, trust the inequalities, and the final answer of $-\sqrt{2}$ will always reveal itself.