Question

Prove each identity. (All identities in this chapter can be proven. ) $$\tan x=\frac{\sec x}{\csc x}$$

Short answer

\(\tan x = \frac{\sec x}{\csc x}\) is proven to be true by recognizing that both sides of the identity simplify to \(\frac{\sin x}{\cos x}\).

Step-by-step solution

Understand the Tangent Identity

Recall that the tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side, so in terms of sine and cosine, \(\tan x = \frac{\sin x}{\cos x}\).

Define Secant and Cosecant

The secant and cosecant are reciprocals of cosine and sine respectively. That means \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\).

Write Tangent in Terms of Secant and Cosecant

Using the definitions from Step 2, write tangent as \(\tan x = \frac{\sec x}{\csc x}\) so the right side becomes \(\frac{1/\cos x}{1/\sin x}\) which simplifies to \(\frac{\sin x}{\cos x}\), demonstrating that \(\tan x = \frac{\sec x}{\csc x}\) is indeed an identity.

Key concepts

Tangent Identity

The tangent identity is a fundamental concept in trigonometry that relates the tangent of an angle to the other trigonometric functions. Understanding this identity is crucial as it allows for simplifying complex trigonometric equations and solving various mathematical problems. In essence, the tangent of a given angle, say 'x', is defined as the ratio of the sine of 'x' to the cosine of 'x'. Written mathematically, we express the tangent identity as:
\[\tan x = \frac{\sin x}{\cos x}\]
Remember that 'sin' represents the sine function, and 'cos' stands for the cosine function. These functions relate to the sides of a right-angled triangle, where the sine of an angle represents the ratio of the length of the opposite side to the hypotenuse, while cosine relates the length of the adjacent side to the hypotenuse. This understanding helps us comprehend why \[\tan x = \frac{\sec x}{\csc x}\] is a valid identity, as we're essentially comparing two equivalent ratios.

Secant and Cosecant

Secant and cosecant are two of the lesser-used trigonometric functions, but they're equally important when dealing with trigonometric identities. These functions are the reciprocals of the cosine and sine functions, respectively.
Specifically:
\[\sec x = \frac{1}{\cos x}\]
and
\[\csc x = \frac{1}{\sin x}\]
What this means is that whenever we have a trigonometric expression involving secant or cosecant, we can rewrite it in terms of sine and cosine, which are often more intuitive to work with. For example, in our exercise, the expression \[\tan x = \frac{\sec x}{\csc x}\] becomes much clearer when we replace secant and cosecant with their definitions. It also emphasizes the interconnectedness of all the trigonometric functions, making it easier to prove various identities.

Trigonometric Functions

Trigonometric functions form the bedrock of trigonometry, a branch of mathematics that deals with the relationships between angles and sides of triangles, particularly right-angled triangles. The primary trigonometric functions include sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Understanding Relationships

Each of these functions has a specific relationship with an angle in a right triangle. For instance, the sine function relates an angle to the ratio of the opposite side over the hypotenuse, and the cosine function relates to the ratio of the adjacent side over the hypotenuse. The relationships between these functions are not coincidental but are deeply rooted in the geometry of a circle known as the unit circle.

Practical Usage

In the practical world, trigonometric functions are used to model periodic phenomena, such as sound waves, light waves, and the motion of pendulums. The study and understanding of these functions enable advancements in various fields including engineering, physics, and even finance. An adeptness with trigonometric identities, like the one explored in this exercise, is an essential skill for students, as it enables the manipulation and simplification of complex equations and enhances problem-solving capabilities.