Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\sin (\theta / 2)}{\csc (\theta / 2)}+\frac{\cos (\theta / 2)}{\sec (\theta / 2)}=1$$

Step-by-Step Solution

Answer

The expression simplifies to 1, verifying the identity.

Step 1: Rewrite fractions using trigonometric identities

Recall that the cosecant function is the reciprocal of the sine function, and the secant function is the reciprocal of the cosine function. Hence, we can write:\[\csc(\theta / 2) = \frac{1}{\sin(\theta / 2)}\] and \[\sec(\theta / 2) = \frac{1}{\cos(\theta / 2)}\].Substituting these identities into the original expression, we get:\[\frac{\sin(\theta / 2)}{\frac{1}{\sin(\theta / 2)}} + \frac{\cos(\theta / 2)}{\frac{1}{\cos(\theta / 2)}}\].

Step 2: Simplify the fractions

Simplify each fraction by multiplying by the reciprocal:\[\sin(\theta / 2) \times \sin(\theta / 2) + \cos(\theta / 2) \times \cos(\theta / 2)\].This simplifies to:\[\sin^2(\theta / 2) + \cos^2(\theta / 2)\].

Step 3: Apply the Pythagorean identity

Recall the fundamental Pythagorean identity:\[\sin^2(x) + \cos^2(x) = 1\].Applying this identity to our expression \(\sin^2(\theta / 2) + \cos^2(\theta / 2)\), we find:\[1\].

Key Concepts

Cosecant Function

The cosecant function, abbreviated as \( \csc \), is one of the basic trigonometric functions. It is often described in relation to the sine function. Specifically, the cosecant is the reciprocal of sine. This means for an angle \( \theta \), if the sine of \( \theta \) is \( \sin(\theta) \), then the cosecant of \( \theta \) is \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

• This reciprocal nature makes the cosecant undefined wherever the sine is zero.
• The cosecant function is useful in solving trigonometric equations, especially those involving right triangles.

For example, in the given exercise, the expression \( \frac{\sin(\theta / 2)}{\csc(\theta / 2)} \) can be simplified by recognizing \( \csc(\theta / 2) \) as \( \frac{1}{\sin(\theta / 2)} \).
This highlights the power of understanding reciprocal relationships in trigonometry. Understanding the cosecant function provides a deeper insight into the trigonometric identities and aids in solving more complex problems.

Secant Function

Moving on to the secant function, it is closely related to the cosine function, much like the relationship between cosecant and sine. The secant function, denoted as \( \sec \), is defined as the reciprocal of cosine. That means \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

• Similar to cosecant, the secant function is undefined wherever the cosine function is zero.
• It is essential in trigonometry, often used in equations or identities involving angles.

In the context of the exercise, recognizing the secant at \( \theta/2 \) helps simplify expressions such as \( \frac{\cos(\theta / 2)}{\sec(\theta / 2)} \) into \( \cos(\theta / 2) \times \cos(\theta / 2) \).
Recognizing these reciprocal functions is important as it simplifies the expressions effectively. With a strong grasp of the secant, much like the cosecant, one can easily handle many trigonometric identities and equations.

Pythagorean Identity

One of the most fundamental identities in trigonometry is the Pythagorean identity. It forms the backbone for many seven trigonometric problems. The key formula is \( \sin^2(x) + \cos^2(x) = 1 \), which emerges from the Pythagorean theorem applied to a unit circle.

• This identity holds for any angle \( x \) and is a cornerstone in simplifying trigonometric expressions.
• Understanding this identity allows you to transform complex expressions and verify identities.

In our exercise, after substituting and simplifying using reciprocal identities, the expression \( \sin^2(\theta / 2) + \cos^2(\theta / 2) \) can be directly evaluated using the Pythagorean identity to equal \( 1 \).
This demonstrates how powerful the Pythagorean identity is in transforming and verifying trigonometric expressions. Grasping this identity is key to mastering various trigonometric challenges with confidence and efficiency.