Problem 69 - Calculus for Biology and Medicine

Question

Show that the identity $$ 1+\tan ^{2} \theta=\sec ^{2} \theta $$ follows from $$ \sin ^{2} \theta+\cos ^{2} \theta=1 $$

Step-by-Step Solution

Answer

The identity \( 1 + \tan^2 \theta = \sec^2 \theta \) is derived by dividing the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) by \( \cos^2 \theta \).

Step 1: Recall Pythagorean Identity

We start with the well-known Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This is a fundamental trigonometric identity that relates the squares of sine and cosine of the same angle \( \theta \).

Step 2: Divide Both Sides by \( \cos^2 \theta \)

We divide each term in the Pythagorean identity by \( \cos^2 \theta \):\[ \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} \]This simplifies the equation to another common trigonometric identity.

Step 3: Simplify Using Trigonometric Definitions

Recognize that \( \frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta \) and \( \frac{1}{\cos^2 \theta} = \sec^2 \theta \). Therefore, the equation simplifies to:\[ \tan^2 \theta + 1 = \sec^2 \theta \]Thus, we have shown that \( 1 + \tan^2 \theta = \sec^2 \theta \) follows from the original Pythagorean identity.

Key Concepts

Pythagorean Identity

One of the core concepts in trigonometry is the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is derived from the Pythagorean theorem in a right triangle, where the sine and cosine functions represent the ratios of the sides of the triangle. To understand this, think about a right triangle with angles \( \theta \, (90^\circ - \theta) \, and 90^\circ \). Here, \( \sin(\theta) \) represents the opposite side divided by the hypotenuse, and \( \cos(\theta) \) represents the adjacent side divided by the hypotenuse.
By squaring both the sine and cosine and adding them together, you'll always get one, as you're effectively describing the entire length of the hypotenuse (considered as 1 in a unit circle). The importance of the Pythagorean identity is vast in trigonometry, as it forms the basis for many other trigonometric identities and relationships.

Tangent and Secant Relationship

In trigonometry, tangent and secant are two important functions with a unique relationship that can be derived from the Pythagorean identity. To bridge \( \sin^2 \theta + \cos^2 \theta = 1 \) to \( 1 + \tan^2 \theta = \sec^2 \theta \), we need to introduce tangent and secant functions:\

• The tangent function \( \tan(\theta) \) is defined as the ratio of sine to cosine: \( \frac{\sin(\theta)}{\cos(\theta)} \).
• The secant function \( \sec(\theta) \) is the reciprocal of the cosine function: \( \frac{1}{\cos(\theta)} \).

To derive this identity, dividing the whole Pythagorean identity by \( \cos^2 \theta \) transforms it using these definitions:
\[ \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} \]
This step gives rise to \( \tan^2 \theta \) and \( \sec^2 \theta \), confirming that \( 1 + \tan^2 \theta = \sec^2 \theta \). This relationship is particularly useful when dealing with right angle triangles and helps simplify complex trigonometric expressions.

Trigonometric Functions Simplification

Simplifying trigonometric functions is essential in solving equations and proving identities. The goal is often to rewrite expressions into simpler forms using identities and known relationships. In the given exercise, you make use of simplification by dividing each term of the Pythagorean identity by \( \cos^2 \theta \). This tactic allows us to transform a well-known identity into a new form involving tangent and secant.
When simplifying trigonometric functions, it's important to be mindful of core identities like:

• Reciprocal identities such as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Quotient identities like \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• Pythagorean identities which can be manipulated, as seen above.

By practicing the simplification of functions, you'll become more adept at working with trigonometric expressions and better equipped to tackle higher-level problems that involve trigonometry. Remember that each simplification technique provides a valuable tool in your mathematical toolkit.