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Understanding double-angle formulas is vital when working with trigonometric identities because they allow us to express trigonometric functions of double angles in terms of single angles.

Double-angle formulas state that for any angle \( \theta \):

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \), which can also be written as \( 2\cos^2(\theta) - 1 \) or \( 1 - 2\sin^2(\theta) \)
• \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \)

In our exercise, using the double-angle formula \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) is the first step to verify the given identity. This replacement is crucial as it reduces the complexity of the expression and allows further simplifications.

The process of simplifying trigonometric expressions involves various algebraic manipulations to rewrite expressions into a preferred or more recognizable form. This can include expanding products, combining like terms, or converting between trigonometric functions using identities.

When simplifying, one effective strategy is to convert all trigonometric functions to sine and cosine, as these are the basic trigonometric functions from which others are derived. This is exactly what is done in step 2 of our example by expressing cotangent in terms of cosine and sine. Furthermore, recognizing common trigonometric identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \), also aids in simplifying expressions effectively.

The cotangent function, represented as \( \cot \), is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function and can be expressed as the ratio of the adjacent side to the opposite side in a right-angled triangle, or as the ratio of cosine to sine:

\( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} \).

Understanding Cotangent in Trigonometric Identities

When working with trigonometric identities like in our exercise, converting the cotangent function to a ratio of cosine and sine can facilitate the simplification process. After expressing cotangent in terms of cosine and sine, algebraic simplification is possible, especially when combined with other trigonometric identities to reduce the complexity of the expression.