Problem 28 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\tan x=\sqrt{\sec ^{2} x-1}$$

Step-by-Step Solution

Answer

In conclusion, the given equation, \(\tan x = \sqrt{\sec^2 x - 1}\), is an identity as it holds true for all values of x in its domain. We demonstrated this by analyzing the relationship between \(\tan x\) and \(\sec x\), simplifying the equation, and using the Pythagorean identity.

Step 1: Rewrite the equation using the relationship between \(\tan x\) and \(\sec x\)

We know that \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). This relationship can help us start analyzing the equation better. We can rewrite the given equation as: $$\frac{\sin x}{\cos x} = \sqrt{\sec^2 x - 1}$$ Now, let's replace \(\sec x\) with its equivalent in terms of \(\cos x\): $$\frac{\sin x}{\cos x} = \sqrt{\left(\frac{1}{\cos^2 x}\right) - 1}$$

Step 2: Simplify the equation

Now we need to simplify the equation further. To do this, we can notice that we have a common denominator inside the square root. We can rewrite the original equation as follows: $$\frac{\sin x}{\cos x} = \sqrt{\frac{1 - \cos^2 x}{\cos^2 x}}$$

Step 3: Use the Pythagorean identity

Next, we can use one of the most fundamental trigonometric identities, the Pythagorean identity, which states that \(\sin^2 x + \cos^2 x = 1\). From this, we know that \(\sin^2 x = 1 - \cos^2 x\). Now we can replace the numerator inside the square root with \(\sin^2 x\): $$\frac{\sin x}{\cos x} = \sqrt{\frac{\sin^2 x}{\cos^2 x}}$$

Step 4: Simplify the square root

Now we can simplify the square root further by taking the square root of the fraction: $$\frac{\sin x}{\cos x} = \frac{\sqrt{\sin^2 x}}{\sqrt{\cos^2 x}}$$ Since the square root of a square is just the original value (considering positive values for trigonometric functions in this case): $$\frac{\sin x}{\cos x} = \frac{\sin x}{\cos x}$$

Step 5: Conclusion

By analyzing and simplifying the original equation, we can deduce that the left and right sides of the equation are equal for all possible values of x. Therefore, the given equation is an identity: $$\tan x = \sqrt{\sec^2 x - 1}$$

Key Concepts

Trigonometric Functions

Trigonometric functions are a key cornerstone in mathematics that express relationships between angles and sides of triangles. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each of these functions offers a unique way of interpreting an angle's relation to a circle or triangle. Here’s a quick overview of these functions:

• Sine (\( \sin \)) relates the opposite side of a right triangle to the hypotenuse.
• Cosine (\( \cos \)) accounts for the adjacent side over the hypotenuse.
• Tangent (\( \tan \)) is the ratio of the sine and cosine, \( \tan x = \frac{\sin x}{\cos x} \).

Other related functions that are tremendously useful include secant (\( \sec \)), which is the reciprocal of cosine, or \( \sec x = \frac{1}{\cos x} \). Each function provides unique insights and facilitates the solving of complex mathematical equations and problems through these ratios, making them instrumental in calculations and proofs involving angles and measurements.

Pythagorean Identity

The Pythagorean Identity is one of the fundamental relations in trigonometry. It is derived from the Pythagorean theorem applied to trigonometric functions and is written as \( \sin^2 x + \cos^2 x = 1 \). This identity assists in transitioning between sine and cosine forms, especially when solving trigonometric equations. The relevance of this identity in trigonometry lies in its ability to simplify expressions and demonstrate equivalencies between different trigonometric functions. Here’s how it works:

• By rearranging \( \sin^2 x + \cos^2 x = 1 \) to \( \sin^2 x = 1 - \cos^2 x \), we can express sine in terms of cosine.
• It allows conversion and simplification of trigonometric expressions fitting this pattern.

In our exercise, this identity was pivotal in simplifying the equation by expressing \( \sin^2 x \) in terms of \( \cos^2 x \), leading to a simpler and direct proof of the identity we were asked to examine.

Simplifying Expressions

Simplifying trigonometric expressions involves several techniques that make complex equations easier to manage and solve. It typically starts with recognizing forms and relationships between trigonometric functions, then employing algebraic manipulation to simplify. A step-by-step approach includes:

• Substituting equivalent expressions using trigonometric identities, such as replacing \( \sec x \) with \( \frac{1}{\cos x} \).
• Combining like terms and reducing complex fractions into simpler forms.
• Utilizing identities like the Pythagorean Identity to express terms as simpler equivalent forms.

In this problem, we started by expressing \( \sec^2 x - 1 \) under a square root, breaking it down into a more recognizable trigonometric relationship using our identities. The simplification process is essential because it transforms complex equations into simple identities or expressions that are easier to understand and prove, just like the identity \( \tan x = \sqrt{\sec^2 x - 1} \) that we proved in the exercise.