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The secant function is an important trigonometric identity often abbreviated as "sec". It is directly related to the cosine function and helps create connections between other trigonometric identities. Secant is defined as the reciprocal of the cosine function. Therefore, we can say that:

• Secant of an angle \( x \) is \( \sec x = \frac{1}{\cos x} \).

This means that wherever you encounter the secant function, you can think of it in terms of cosine.
Understanding secant as \( \frac{1}{\cos x} \) allows us to transform problems into equivalent expressions using sine and cosine, which often makes the math easier to work with. In our exercise, we used this identity to rewrite the equation and help simplify both sides. This substitution plays a crucial role in verifying and proving trigonometric identities.

Tangent, often abbreviated as "tan", is a fundamental trigonometric function that can be expressed in terms of sine and cosine. The tangent of an angle \( x \) is given by the ratio of the sine to the cosine of that angle.

• Tangent of \( x \) is \( \tan x = \frac{\sin x}{\cos x} \).

This expression is central in transforming complex trigonometric problems into manageable forms.
By expressing tangent in terms of sine and cosine, we were able to combine it neatly with the secant function in the solution. This blending of fractions makes verifying trigonometric identities more straightforward, as it often brings us down to simpler terms already known and easy to handle.
In this exercise, rewriting \( \tan x \) as \( \frac{\sin x}{\cos x} \) was pivotal for bringing the identity into a form that matched the right side of the expression provided.

The sine and cosine functions are the cornerstones of trigonometry. They describe the relationships between the angles and sides of right triangles, and they're also foundational to the circle definitions of trigonometry.
Sine, abbreviated as "sin", is defined for an angle \( x \) on the unit circle as the y-coordinate of the point where the terminal side of the angle intersects the circle.
Cosine, abbreviated as "cos", similarly represents the x-coordinate at that intersection.

• For a right triangle, \( \sin x = \frac{\text{opposite side}}{\text{hypotenuse}} \).
• And \( \cos x = \frac{\text{adjacent side}}{\text{hypotenuse}} \).

In terms of our exercise, these functions allowed us to express secant and tangent effectively.
When dealing with identities, expressing functions in terms of sine and cosine not only unlocks simpler manipulation but also provides a visual or geometric understanding of the relationships at play.
Thus, understanding sine and cosine lays down the groundwork for simplifying and proving many trigonometric identities, including the one in our exercise.