Problem 39 - Algebra and Trigonometry

Question

Verify each identity. \(\tan ^{2} 2 x+\sin ^{2} 2 x+\cos ^{2} 2 x=\sec ^{2} 2 x\)

Step-by-Step Solution

Answer

Verified, the equation \(\tan ^{2} 2 x+\sin ^{2} 2 x+\cos ^{2} 2 x=\sec ^{2} 2 x\) holds true.

Step 1: Recognize Basic Identities

Recall and use the basic Pythagorean identity in trigonometry, \(\sin^2x + \cos^2x = 1\). The different forms of this relationship represent the fundamental identities in trigonometry, from which the trigonometric functions are defined.

Step 2: Use the Trigonometric Identity

Remember that \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\), which allows us to write \(\sec^2 x = 1 + \tan^2 x\). Applying this on the left side of the equation to match the form on the right side, the original equation transforms into \(\tan^2 2x + 2 = \sec^2 2x\).

Step 3: Verify the Identity

Subtract \(\tan^22x\) from both sides of the equation to get \(\sec^22x - \tan^22x = 2\), which verifies our given identity.

Key Concepts

Pythagorean identity

The Pythagorean identity is fundamental in trigonometry and states that for any angle \( x \), the sum of the squares of the sine and cosine functions is always 1: \( \sin^2 x + \cos^2 x = 1 \). This identity forms the cornerstone for many other trigonometric identities because it establishes a basic relationship between sine and cosine. When you're dealing with more complex trigonometric identities, always look for opportunities to apply this identity.
In the context of verifying other identities, such as the problem at hand, this identity helps simplify expressions, allowing you to express one trigonometric function in terms of another. This makes it easier to prove or refute the given equation.

secant function

The secant function, denoted by \( \sec x \), is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). It is defined wherever cosine is non-zero, which means you cannot have a secant value where \( \cos x = 0 \). This reciprocal relationship is crucial because it transforms complex equations into simpler forms.

• It is especially useful when verifying identities because it can change division into multiplication.
• Knowing that \( \sec^2 x = 1 + \tan^2 x \) helps bridge relationships between tangent and secant in complex identities.

In the given exercise, by identifying the relationship between secant and tangent, we can transform and simplify the specified identity to verify it more easily.

tangent function

The tangent function, represented as \( \tan x \), is the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \). This introduces an important aspect of trigonometry where tangent can help connect sine and cosine, useful in both practical applications and theoretical proofs.

• The squared tangent, \( \tan^2 x \), often shows up in identities due similar proportions that sine and cosine share.
• Using the identity \( \sec^2 x = 1 + \tan^2 x \) allows us to express the secant function in terms of tangent, further simplifying the verification of identities that involve squares.

In solving our exercise, recognizing how tangent relates to these functions was integral to rewriting and verifying the identity.

algebraic verification

Algebraic verification involves proving mathematical identities or equations by transforming one side of the equation to match the other using algebraic manipulations or known identities. This step-by-step conversion relies on logical reasoning and a solid understanding of the rules of algebra and trigonometry.
In the exercise given, transformation started by recalling that \( \sec^2 x = 1 + \tan^2 x \). By substituting these values into the equation and simplifying the expression, one attempts to reach an equivalent expression on both sides of the equation.

• We verify the identity by converting the left-hand side into a form that matches the right-hand side.
• Subtracting terms on either side of the equation is a typical strategy to isolate and verify components.

Ultimately, verification boils down to ensuring both sides of the equation represent the same value under all conditions defined by the initial identity.