Problem 23 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\tan ^{4} k-\sec ^{4} k=1-2 \sec ^{2} k$$

Step-by-Step Solution

Answer

The identity is verified: \( \tan^4 k - \sec^4 k = 1 - 2\sec^2 k \).

Step 1: Express tangent and secant in terms of sine and cosine

Recall the definitions of \(\tan k\) and \(\sec k\):\[ \tan k = \frac{\sin k}{\cos k} \quad \text{and} \quad \sec k = \frac{1}{\cos k} \] Using these definitions, we can express \(\tan^2 k\) and \(\sec^2 k\):\[ \tan^2 k = \frac{\sin^2 k}{\cos^2 k} \] \[ \sec^2 k = \frac{1}{\cos^2 k} \]

Step 2: Apply the Pythagorean identity

By using the Pythagorean identity \(1 + \tan^2 k = \sec^2 k\), substitute \(\tan^2 k\) with \(\sec^2 k - 1\):\[ \tan^4 k = (\tan^2 k)^2 = (\sec^2 k - 1)^2 \] Thus,\[ \tan^4 k = (\sec^2 k - 1)(\sec^2 k - 1) = \sec^4 k - 2\sec^2 k + 1 \]

Step 3: Substitute and simplify the left-hand side expression

Now substitute \(\tan^4 k + \sec^4 k\) using the simplification from the previous step:\[ \tan^4 k - \sec^4 k = (\sec^4 k - 2\sec^2 k + 1) - \sec^4 k \] Combine the terms:\[ (\sec^4 k - \sec^4 k) - 2\sec^2 k + 1 = 1 - 2 \sec^2 k \]

Step 4: Compare the simplified expression with the right-hand side

Observe that the simplified expression from the left side of the identity, i.e., \(1 - 2\sec^2 k\), is exactly the same as the right-hand side of the original identity. Since both sides match, the identity is verified.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the cornerstones of trigonometry. It states that for any angle \( k \), the following equation holds: \[ 1 + an^2 k = \sec^2 k \] This identity is derived from the well-known Pythagorean theorem in a right-angled triangle, connecting the tangent and secant functions. Recalling this identity is fundamental, as it helps simplify complex trigonometric expressions in problems where functions like tangent and secant make an appearance. Notice how it makes possible the substitution of \( an^2 k \) with \( \sec^2 k - 1 \), facilitating the simplification process.

Tangent Function

The Tangent Function, \( \tan k \), is a basic trigonometric function defined as the ratio of the opposite to the adjacent side in a right-angled triangle. Expressed in terms of sine and cosine, it looks like this: \[ \tan k = \frac{\sin k}{\cos k} \] When you square the tangent function: \[ \tan^2 k = \frac{\sin^2 k}{\cos^2 k} \] Using this squared form, you can analyze complex expressions more easily, especially when coupled with trigonometric identities. It's crucial to remember that tangent represents the slope of the angle in a unit circle, often aiding in visualizing and verifying trigonometric identities.

Secant Function

The Secant Function, denoted as \( \sec k \), extends the concept of cosine. It is defined as the reciprocal of the cosine function: \[ \sec k = \frac{1}{\cos k} \] Squaring the secant function results in: \[ \sec^2 k = \frac{1}{\cos^2 k} \] The secant function is vital in trigonometry because it simplifies the process of managing expressions involving divisions by cosine. It's an integral component of many identities and formulas, such as the Pythagorean Identity. By considering \( \sec k \), you gain another tool necessary for the simplification of trigonometric expressions.

Trigonometric Simplification

Trigonometric Simplification is a powerful tool used to make complex trigonometric expressions easier to handle. In verifying identities, it often involves expressing functions in terms of sine and cosine or employing known identities such as the Pythagorean Identity. Here is a brief guide to aid your simplification journey:

• Convert expressions to base trigonometric functions like sine and cosine.
• Utilize identities like \( 1 + \tan^2 k = \sec^2 k \) to replace functions and reduce the expression size.
• Combine like terms and simplify expressions step by step, aiming to match the target expression on the other side of an equation.

In our original problem, understanding trigonometric simplification allowed us to transform a quite complex identity to the point where both sides matched precisely, verifying the given identity.