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Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. These identities are crucial for simplifying expressions and solving equations in calculus and other branches of mathematics. One fundamental identity is the Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \). This identity helps in expressing one trigonometric function in terms of another, making it easier to tackle complex problems.
Another key identity is that's directly applicable to our exercise is \( \sin^2\theta = \frac{1 - \cos(2\theta)}{2} \). This relationship helps simplify powers of sine into more manageable expressions. Trigonometric identities are like shortcuts in math that often allow us to transform a tough problem into a simpler one.

The double angle formula is a set of equations used to express trigonometric functions of twice the angle in terms of the trigonometric functions of the original angle. For sine, the formula is \( \sin(2\alpha) = 2 \sin\alpha \cos\alpha \). The double angle formula allows us to break down these trigonometric problems into simpler situations involving smaller angles.
In the given problem, we apply it to rewrite \( \sin 2x \) as \( 2 \sin x \cos x \). This is particularly useful when dealing with trigonometric equations as it transforms a single complex term into a product of two simpler trigonometric functions. This makes it easier to analyze and solve the given equation using basic arithmetic operations.

Equation solving techniques involve methods to find the values of variables satisfying the equality conditions. Here, comparing both sides of the equation becomes the primary technique.
Let's break it down:

• First, simplify both sides using trigonometric identities or formulas, like how we used the double angle formula to transform the equation.
• Ensure all expressions are as simplified as possible, allowing for easier comparison.
• Finally, carefully compare each side. Ideally, any viable solution would balance the equation on both sides.

This is where understanding each transformation and simplification step is critical. Once an equation reaches a form where one side equates to another, or shows an inconsistency, like \( 2 = 0 \), it indicates whether solutions exist or not. In this case, the inconsistency demonstrates that no solutions satisfy the original equation.