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Magazine Chapter 7: Problem 71
Verify the identity. $$\sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x$$
Short Answer
Expert verified
The identity is verified: both sides simplify to \( \sec^2 x + \tan^2 x \).
Step by step solution
01
Understanding the Identity
We need to verify that \( \sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x \). To do this, we will start by rewriting the left side of the equation by factoring it.
02
Factoring the Left Side
Notice that \( \sec^4 x - \tan^4 x \) is a difference of squares: \((\sec^2 x)^2 - (\tan^2 x)^2\). Using the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\), we can factor it as \((\sec^2 x - \tan^2 x)(\sec^2 x + \tan^2 x)\).
03
Simplifying the Factored Expression
The factored expression is \((\sec^2 x - \tan^2 x)(\sec^2 x + \tan^2 x)\). We already see the term \(\sec^2 x + \tan^2 x\) on the right side of the identity. We now need to simplify \(\sec^2 x - \tan^2 x\).
04
Using Trigonometric Identity
Recall the identity \(\sec^2 x = 1 + \tan^2 x\). Thus, we have \(\sec^2 x - \tan^2 x = (1 + \tan^2 x) - \tan^2 x = 1\).
05
Completing the Proof
Since \(\sec^2 x - \tan^2 x = 1\), substituting this into the factored form gives us \((1)(\sec^2 x + \tan^2 x) = \sec^2 x + \tan^2 x\), which matches the right side of the identity we wanted to verify.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Secant Function
The secant function, represented as \( \sec x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function. This means that \( \sec x = \frac{1}{\cos x} \). Because secant is derived from cosine, it shares similar properties regarding its domain and range.
- Domain: The secant function is undefined where \( \cos x = 0 \) because division by zero is undefined. Therefore, \( \sec x \) is undefined at odd multiples of \( \frac{\pi}{2} \).
- Range: The range of \( \sec x \) is \( (-\infty, -1] \cup [1, \infty) \). This is because cosine values range between -1 and 1, leading the reciprocal to be outside these bounds.
Tangent Function
The tangent function, represented by \( \tan x \), is another primary trigonometric function that describes the ratio of the opposite side to the adjacent side in a right triangle. It is defined as \( \tan x = \frac{\sin x}{\cos x} \). Below are some key properties:
- Domain: Tangent is undefined at points where \( \cos x = 0 \), which are odd multiples of \( \frac{\pi}{2} \).
- Range: The range of \( \tan x \) is all real numbers (\( (-\infty, \infty) \)). This is because sine and cosine repeat throughout their cycles, but since tangent is their quotient, it creates a repeating, yet distinct pattern.
- Periodicity: Tangent has a periodicity of \( \pi \). This means that \( \tan(x + \pi) = \tan x \).
Difference of Squares
The difference of squares is a specific algebraic expression of the form \( a^2 - b^2 \). It can be factored using the formula \( a^2 - b^2 = (a-b)(a+b) \). This technique is frequently used in simplifying algebraic expressions, including those found within trigonometry.
- Application: In the exercise, the identity \( \sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x \) was simplified using the difference of squares. Recognizing \( \sec^4 x - \tan^4 x \) as \((\sec^2 x)^2 - (\tan^2 x)^2 \) enabled factorization into \( (\sec^2 x - \tan^2 x)(\sec^2 x + \tan^2 x) \).
- Simplification: This step allowed further simplification using known trigonometric identities. Specifically, \( \sec^2 x - \tan^2 x = 1 \) helped resolve the identity by demonstrating that the left side equals the right side.
