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The double angle identity for cosine is an important concept in trigonometry. It is particularly useful when dealing with angles that are twice the measure of a given angle. The basic form of this identity is: - \( \cos 2x = \cos^2 x - \sin^2 x \). This identity is valuable because it allows us to manipulate trigonometric expressions effectively, like deriving other forms such as tangent. To reframe \( \cos 2x \) in terms of tangent, one might express the double angle identity using known relationships between sine, cosine, and tangent functions. It's also common to express it as a fraction: - \( \cos 2x = \frac{\cos^2 x - \sin^2 x}{\cos^2 x + \sin^2 x} \) Since \( \cos^2 x + \sin^2 x = 1 \), it simplifies, highlighting a versatile piece of the trigonometric puzzle.

Tangent, one of the basic trigonometric functions, is the ratio of sine to cosine. It can be defined as: - \( \tan x = \frac{\sin x}{\cos x} \). This relationship allows us to rewrite expressions involving sine and cosine in terms of tangent. For instance, knowing that \( \sin x = \tan x \cdot \cos x \) lets us find sine if tangent and cosine are known. Also, tangent can be related back to other trigonometric identities easily. This exercise's goal is to express cosine double angle identities using tangent. With tangent, we focus more on simplifying expressions creatively, which aids in understanding trigonometric proofs.

Simplifying trigonometric expressions often requires understanding and applying identities effectively. In this exercise, we express \( \cos 2x \) with tangent to simplify it. - Start by substituting sine and cosine in terms of tangent: - \( \sin x = \tan x \cdot \cos x \) - \( \cos x = \frac{1}{\sqrt{1 + \tan^2 x}} \) Inserting these into the identity for \( \cos 2x \) becomes essential: - \( \cos 2x = \left(\frac{1}{\sqrt{1 + \tan^2 x}} \right)^2 - (\tan x \cdot \cos x)^2 \) This simplifies further to: - \( \cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x} \), thus proving the identity. Simplifying these trigonometric expressions is all about translating complex forms into simpler, recognizable identities.