Animated Solution for Mathematics - Trigonometry: For any $\theta \in (\frac{\pi }{4},\frac{\pi }{2})$, the expression $3(\sin \theta -\cos \theta {)}^4+6(\sin \theta +\cos \theta {)}^2+4{\sin }^6\theta$ equals :

To solve the expression $3(\sin \theta -\cos \theta {)}^4+6(\sin \theta +\cos \theta {)}^2+4{\sin }^6\theta$, we must first identify the underlying trigonometric patterns. The terms $(\sin \theta \pm \cos \theta )$ are classic indicators of the identity:

By recognizing this, we can simplify the powers of these binomials. We treat the fourth power as the square of a square: $[(\sin \theta -\cos \theta {)}^2{]}^2$.

Substituting the identity into our expression, we transform the original problem into:

Now, we expand the squared term $3(1-\sin 2\theta {)}^2$ to get $3(1-2\sin 2\theta +{\sin }^{2}2\theta )$, which simplifies to:

Adding the second term, $6+6\sin 2\theta$, we observe that the linear terms in $\sin 2\theta$ cancel out perfectly:

To reach the final form, we express the remaining terms in terms of $\cos \theta$. Recall that ${\sin }^{2}2\theta =4{\sin }^2\theta {\cos }^2\theta =4(1-{\cos }^2\theta ){\cos }^2\theta$.

Substituting this back into our simplified expression, we perform the algebraic reduction. Through this systematic simplification, the expression resolves to:

This result demonstrates that even the most intimidating trigonometric expressions can be dismantled by identifying core identities and maintaining algebraic discipline.