Problem 73 - Algebra and Trigonometry

Question

Verify the identity. $$ \sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x $$

Step-by-Step Solution

Answer

The identity is verified; both sides are equivalent.

Step 1: Verify Identity Form

We start by examining both sides of the equation \( \sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x \). Our goal is to prove that both sides are equivalent for all values of \(x\).

Step 2: Apply Pythagorean Identity

Use the Pythagorean identity \( \sec^2 x = 1 + \tan^2 x \). Using this identity, we can transform the expression by rewriting \( \sec^4 x \) and \( \tan^4 x \).

Step 3: Express Powers

Rewrite \( \sec^4 x \) as \( (\sec^2 x)^2 \) and \( \tan^4 x \) as \( (\tan^2 x)^2 \). Substitute \( \sec^2 x = 1 + \tan^2 x \) into \( \sec^4 x \) which now becomes \( (1 + \tan^2 x)^2 \).

Step 4: Expand Squares

Expand \((1 + \tan^2 x)^2\) to \(1 + 2\tan^2 x + \tan^4 x\). Thus, the expression \(\sec^4 x - \tan^4 x\) becomes \(1 + 2\tan^2 x + \tan^4 x - \tan^4 x\).

Step 5: Simplify Expression

Simplify \(1 + 2\tan^2 x + \tan^4 x - \tan^4 x\) by canceling \(\tan^4 x\). This reduces to \(1 + 2\tan^2 x\).

Step 6: Simplify Further to Match Right Side

Rewrite \(1 + \tan^2 x + \tan^2 x\) as \(\sec^2 x + \tan^2 x\) using \(\sec^2 x = 1 + \tan^2 x\).

Step 7: Conclusion

Both sides of the original identity equation are equivalent. Therefore, \( \sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x\) is a true identity.

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental relationship in trigonometry that connects the secant and tangent functions. Knowing and applying this identity makes verifying trigonometric identities simpler. The Pythagorean identity states that:

• \( \sec^2 x = 1 + \tan^2 x \)

This equation arises from the Pythagorean theorem applied to a right triangle. As secant and tangent are related to the cosine and sine functions respectively, understanding this identity is essential when working with different trigonometric expressions.
In the exercise, we use the Pythagorean identity to rewrite complex expressions into simpler forms. For instance, \( \sec^4 x \) can be rewritten as \((\sec^2 x)^2\) and further expressed using \(\sec^2 x = 1 + \tan^2 x\). This powerful strategy allows for manipulating terms and verifying identities more easily.

trigonometric proofs

Trigonometric proofs often involve showing that two expressions are equivalent. This task is simplified by skillfully using identities like the Pythagorean identity. Proofs are not simply about plugging in values; they require creative problem-solving and logical steps.
In our exercise, the primary objective was to verify the trigonometric identity \( \sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x \). Proofs usually start by transforming one side of an equation to resemble the other. Steps include:

• Rewriting expressions using fundamental identities, such as the Pythagorean identity.
• Manipulating algebraic expressions, such as expanding or factoring.
• Logical simplifications to reveal equivalent forms.

By understanding the methodical process of a proof, students can more confidently solve trigonometric problems and deepen their grasp of the underlying concepts.

secant and tangent functions

Secant and tangent are two of the six primary trigonometric functions, with distinct definitions related to angles and ratios in a right triangle.

• The secant function \( \sec x \) is defined as \( \frac{1}{\cos x} \).
• The tangent function \( \tan x \) is defined as \( \frac{\sin x}{\cos x} \).

Secant and tangent are useful for various applications in trigonometry, physics, and engineering. They can appear at first to be somewhat abstract, but the Pythagorean identity bridges their relationship, offering a clearer understanding.
In the exercise, \( \sec^4 x \) and \( \tan^4 x \) were manipulated and simplified using their mathematical definitions and identities. Recognizing how these functions relate through identities helps efficiently simplify and solve trigonometric expressions.