Problem 23 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{1-\cos \theta}{\sin \theta}=\csc \theta-\cot \theta\)

Step-by-Step Solution

Answer

After substituting the identities \( \csc \theta \) and \( \cot \theta \) with their respective expressions in terms of sin and cos, and simplifying the RHS of the equation, it is found that the LHS of the given equation is indeed equal to the RHS, hence verifying the identity.

Step 1: Recognizing the Identities

The first step is identifying the identities \( \csc \theta \) and \( \cot \theta \) . Both identities can be expressed in terms of sin and cos. \( \csc \theta \) is equal to \( \frac{1}{\sin \theta} \) and \( \cot \theta \) is equal to \( \frac{\cos \theta}{\sin \theta} \) .

Step 2: Substituting the Identities in the Right Hand Side (RHS)

Now, substitute these expressions into the RHS of the equation. This gives \( \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} \) .

Step 3: Simplifying the RHS

After substituting, look for like terms to combine them. The two fractions have a common denominator of \( \sin \theta \) . By combining the fractions over this common denominator, the RHS equation simplifies to \( \frac{1-\cos \theta}{\sin \theta} \) .

Step 4: Comparing the LHS and the RHS

Now, compare the LHS with the RHS. As it can be seen, the LHS \( \frac{1-\cos \theta}{\sin \theta} \) is equal to the RHS \( \frac{1-\cos \theta}{\sin \theta} \) . Hence, the identity is verified.

Key Concepts

Reciprocal Identities

Reciprocal identities are a fundamental part of trigonometry, especially when dealing with the fundamental trigonometric functions. They help in transforming one trigonometric function into another, making it easier to simplify or solve equations. The three basic reciprocal identities are:

• The cosecant function, which is the reciprocal of sine, \( \csc \theta = \frac{1}{\sin \theta} \).
• The secant function, reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \).
• The cotangent function, reciprocal of tangent, \( \cot \theta = \frac{1}{\tan \theta} \).

These identities are vital for verifying trigonometric equations because they often allow for simplification. By converting terms into their reciprocal forms, equations that initially seem complicated can often be simplified into more manageable expressions. Understanding reciprocal identities not only helps in verification but is also crucial when solving complex trigonometric problems.

Cosecant Function

The cosecant function is one of the lesser-known trigonometric functions, but it plays a critical role in many applications. It is symbolized as \( \csc \theta \) and is defined as the reciprocal of the sine function. Specifically, \[ \csc \theta = \frac{1}{\sin \theta} \]This can be very useful when dealing with equations that include sine, particularly when \( \sin \theta eq 0 \). By converting sine to cosecant, you often manage to streamline calculations or verify trigonometric identities. It's also important to remember that the cosecant function is undefined wherever the sine function equals zero because division by zero is undefined. This means that at angles like 0°, 180°, 360°, etc., the cosecant function cannot be calculated. Conversely, wherever sine reaches its maximum (i.e., \( \sin \theta = 1 \) or \( \sin \theta = -1 \)), the value of cosecant will simply be 1 or -1.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is another reciprocal trigonometric function, closely related to tangent. It is defined as the reciprocal of the tangent function. Therefore, \( \cot \theta = \frac{1}{\tan \theta} \), or in terms of sine and cosine, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).This definition is particularly useful for trigonometric simplification and transformations. By rewriting tangent as cotangent, equations can often be reduced to simpler forms that are more easy to deal with.It is important to remember where the cotangent function might be undefined. Since cotangent involves division by sine, it is undefined wherever \( \sin \theta = 0 \). Just like the cosecant, this will happen at angles such as 0°, 180°, and 360°, ensuring that like many trigonometric functions, cotangent has its limitations.Overall, understanding how cotangent relates to the other trigonometric functions is incredibly beneficial, especially in verifying identities or solving trigonometric equations.