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 \(\sin^2\theta+\cos^2\theta=1\) by \(\cos^2\theta\).

Step 1: Start with known Pythagorean identity

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

Step 2: Divide through by \(\cos^2 \theta\)

To make use of the Pythagorean identity to show \(1 + \tan^2 \theta = \sec^2 \theta\), we divide both sides of the identity \(\sin^2 \theta + \cos^2 \theta = 1\) by \(\cos^2 \theta\). This yields: \[ \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} \].

Step 3: Simplify the expression

Simplify each term: The term \( \frac{\sin^2 \theta}{\cos^2 \theta} \) simplifies to \( \tan^2 \theta \), the term \( \frac{\cos^2 \theta}{\cos^2 \theta} \) simplifies to 1, and the term \( \frac{1}{\cos^2 \theta} \) simplifies to \( \sec^2 \theta \). Therefore, the equation simplifies to \( \tan^2 \theta + 1 = \sec^2 \theta \).

Step 4: Conclusion

We have derived the identity \( 1 + \tan^2 \theta = \sec^2 \theta \) starting from the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). This completes the proof.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric relation that states:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. It's crucial because it forms the basis for many other trigonometric identities.
By envisioning a right triangle inscribed in a unit circle, the sine and cosine of an angle \(\theta\) can be interpreted as the lengths of the opposite and adjacent sides.
This inherent relationship always holds true regardless of the angle, making the Pythagorean identity highly reliable.
In proving other identities, such as the identity involving tangent and secant, the Pythagorean identity often acts as the starting point.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the basic trigonometric functions. It is defined as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This means the tangent of an angle is the ratio of the sine of the angle to the cosine of the angle.
This function has interesting properties:

• If \(\theta\) equals angles like \( 0, \pi, 2\pi, \ldots \), then \( \tan \theta = 0 \).
• If \(\theta\) equals odd multiples of \(\frac{\pi}{2}\), the function is undefined due to division by zero in the cosine.
• The tangent function has a repeating period of \( \pi \), making it useful in periodic phenomena.

Understanding tangent as a ratio helps in simplifying and proving identities, such as \(1 + \tan^2 \theta = \sec^2 \theta\).

Secant Function

The secant function is another important trigonometric function, expressed as \( \sec \theta \), and is defined by:

• \( \sec \theta = \frac{1}{\cos \theta} \)

The secant is essentially the reciprocal of the cosine function.
Like other trigonometric functions, it also has distinct characteristics:

• The secant is undefined for angles where the cosine is zero (e.g., \(\theta = \pi/2, 3\pi/2,\ldots \)).
• Its period repeats every \(2\pi\), much like the cosine function from which it is derived.
• \( \sec \theta \) only exists for angles with a non-zero cosine.

The use of the secant function is vital in transforming expressions, particularly when used alongside the tangent function as seen in the identity \(1 + \tan^2 \theta = \sec^2 \theta\). By recognizing how \( \sec \theta \) functions as a reciprocal, we can relate various trigonometric properties effectively.