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Sum-to-product identities are essential tools in trigonometry that allow us to transform sums or differences of sines and cosines into products. This simplifies complex expressions and is particularly helpful when verifying trigonometric identities. When we encounter an expression like \( \sin x \pm \sin y \), the sum-to-product identity says

• \( \sin x \pm \sin y = 2\sin \frac{x \pm y}{2} \cos \frac{x \mp y}{2} \)

For \( \cos x + \cos y \), the identity is

• \( \cos x + \cos y = 2\cos \frac{x + y}{2} \cos \frac{x - y}{2} \)

These identities express the sine or cosine of two angles as a product of sines and cosines of half angles. It can simplify mathematical expressions and help bridge the gap between different trigonometric formats. By applying these identities, fractions can sometimes simplify to forms that enable further transformation or cancellation, such as the situation in the original exercise.

Double-angle identities are another fundamental set of equations in trigonometry that allow us to express trigonometric functions of double angles in terms of single angles. One widely used double-angle identity involves the cosine function:

• \( \cos 2\theta = 2\cos^2 \theta - 1 \)

We can rearrange this to find

• \( \cos \theta = \sqrt{\frac{1+\cos 2\theta}{2}} \)

This identity helps to transform or verify trigonometric expressions, as in our exercise when simplifying the expression by expressing terms like \( \cos \frac{x \mp y}{2} \) in terms of cosine of a single angle. Knowledge of these identities makes solving trigonometric equations more straightforward by providing alternative ways to express complex angles, making them invaluable in trigonometry.

In trigonometry, the tangent function is one of the basic trigonometric functions. It can be defined as the ratio of the sine of an angle to the cosine of the same angle. Mathematically, this is expressed as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This relationship is pivotal when verifying identities like the one in the exercise. Here, the left-hand side of the equation simplifies into the tangent form because, by definition, dividing \( \sin \theta \) by an expression involving \( \cos \theta \), as we did, yields \( \tan \theta \). Understanding the tangent's definition allows for recognizing when this transformation is possible, simplifying equations significantly and ensuring the left-hand side matches the right-hand side or alternate expressions of the identity.