Problem 31 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{\sec ^{4} t-\tan ^{4} t}{1+2 \tan ^{2} t}=1 $$

Step-by-Step Solution

Answer

The identity is verified since both sides simplify to 1.

Step 1: Rewrite Identity using Trigonometric Identities

To begin solving the problem, recall the Pythagorean identity: \( \sec^2 t = 1 + \tan^2 t \). We aim to simplify the given identity: \( \frac{\sec^4 t - \tan^4 t}{1 + 2\tan^2 t} = 1 \). Start by expressing \( \sec^4 t \) in terms of \( \tan^2 t \). We know: \( \sec^2 t = 1 + \tan^2 t \), so \( \sec^4 t = (\sec^2 t)^2 = (1 + \tan^2 t)^2 \).

Step 2: Simplify \( \sec^4 t - \tan^4 t \)

Now, we simplify \( \sec^4 t - \tan^4 t \) using the identity we established. Notice that:\[ \sec^4 t - \tan^4 t = (\sec^2 t + \tan^2 t)(\sec^2 t - \tan^2 t) \].Since \( \sec^2 t = 1 + \tan^2 t \), \( \sec^2 t - \tan^2 t = 1 \). Thus:\[ (\sec^2 t + \tan^2 t)(1) = \sec^2 t + \tan^2 t = 1 + \tan^2 t + \tan^2 t = 1 + 2\tan^2 t \].

Step 3: Verify the Entire Identity

Substitute the simplified forms back into the left-hand side of the identity:\[ \frac{\sec^4 t - \tan^4 t}{1 + 2\tan^2 t} = \frac{1 + 2\tan^2 t}{1 + 2\tan^2 t} \].This simplifies to \(1\), confirming that the left-hand side equals the right-hand side of the original identity. Therefore, the identity holds.

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the square of the secant and tangent functions. This identity states that \( \sec^2 t = 1 + \tan^2 t \). It's derived from the Pythagorean theorem, which relates the sides of a right triangle. By expanding this relationship into trigonometric functions, the identity provides a way to express secant and tangent in terms of each other.

In the context of our exercise, the Pythagorean identity helps us transform \( \sec^4 t \) in terms of \( \tan^2 t \). By recognizing that \( \sec^2 t = 1 + \tan^2 t \), we can square both sides to obtain \( \sec^4 t = (1 + \tan^2 t)^2 \). This transformation is key in verifying the given trigonometric identity.

• Use the identity to simplify complex expressions involving sec and tan.
• Remember that it stems from geometric principles, hence its importance.

simplification of expressions

Simplifying trigonometric expressions is a crucial skill for solving more complex problems. It involves transforming a given expression into a simpler form without changing its value. In our exercise, we aim to simplify \( \sec^4 t - \tan^4 t \) using identities.

One way to simplify this expression is to recognize that it can be factored as a difference of squares: \( \sec^4 t - \tan^4 t = (\sec^2 t + \tan^2 t)(\sec^2 t - \tan^2 t) \). Knowing from the Pythagorean identity that \( \sec^2 t = 1 + \tan^2 t \), the term \( \sec^2 t - \tan^2 t \) simplifies directly to 1.

• Factoring techniques can greatly simplify expressions.
• Leveraging identities can make calculations more straightforward and less prone to error.

By substituting and rearranging terms using identities, the expression \( \sec^4 t - \tan^4 t \) reduces to a form that's straightforward to verify against the given equation.

verifying trigonometric identities

Verifying trigonometric identities involves a process where you show that both sides of an equation are equal by using known identities and simplification techniques. The goal is to manipulate one or both sides of the identity until they are the same.

Our given identity is \( \frac{\sec^4 t - \tan^4 t}{1 + 2 \tan^2 t} = 1 \). By simplifying the numerator \( \sec^4 t - \tan^4 t \) as done before, and noticing that it matches the denominator after simplification, \( 1 + 2 \tan^2 t \), we can see that the expression reduces to \( \frac{1 + 2 \tan^2 t}{1 + 2 \tan^2 t} = 1 \).

• Confirm the equality by transforming expressions methodically.
• Every step should maintain the equality, ensuring the original identity holds true.

This method not only confirms the truth of the expression but also deepens your understanding of the relationships between trigonometric functions.