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Double angle identities are a set of trigonometric expressions that are particularly useful for simplifying calculations involving angles that are twice another angle. These identities allow us to determine the values of trigonometric functions for twice the angle. There are specific formulas for sine, cosine, and tangent, such as:

• For cosine: \(\cos 2x = \cos^2 x - \sin^2 x\) or \(\cos 2x = 2 \cos^2 x - 1\)
• For sine: \(\sin 2x = 2 \sin x \cos x\)
• For tangent: \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\)

In the given problem, using these double angle identities helps in correctly finding the trigonometric functions for \(2x\). By employing these identities step by step, it becomes straightforward to compute \(\cos 2x\), \(\sin 2x\), and \(\tan 2x\). Each identity simplifies the calculations by turning them into algebraic expressions based on known functions.

Trigonometric functions are essential in mathematics, especially when dealing with angles and modeling cycles. These functions include sine, cosine, and tangent, each with unique properties. They are often used to express relations between the angles and sides of triangles. Each function helps to resolve different aspects of a problem:

• The sine function \(\sin x\) represents the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function \(\cos x\) represents the ratio of the adjacent side to the hypotenuse.
• The tangent function \(\tan x\) is defined as the ratio of the sine to the cosine: \(\tan x = \frac{\sin x}{\cos x}\).

Understanding these trigonometric functions and how they relate is key in applying formulas like double angle identities in problems such as the one described.

Quadrant analysis is a crucial step in assessing the properties of trigonometric functions. The Cartesian coordinate system divides the plane into four quadrants, each influencing the sign of trigonometric values differently:

• In the first quadrant, all functions \(\sin x\), \(\cos x\), and \(\tan x\) are positive.
• In the second quadrant, \(\sin x\) is positive, while \(\cos x\) and \(\tan x\) are negative.
• In the third quadrant, both \(\tan x\) and \(\sin x\) are negative, therefore, their quotient \(\tan x\) is positive.
• In the fourth quadrant, \(\cos x\) is positive, while \(\tan x\) and \(\sin x\) are negative.

In the problem, the given range for \(x\) is \(\pi < x < 3\pi/2\), indicating that \(x\) is in the third quadrant. This informs us that \(\cos x\) and \(\sin x\) will be negative, while \(\tan x\) will be positive, guiding us in properly assessing the calculated values of the trigonometric functions for \(2x\). Quadrant analysis ensures no error in the signs of the function values, helping in delivering accurate results.