Problem 69 - Algebra and Trigonometry

Question

Establish the identity: \(\csc \theta-\sin \theta=\cos \theta \cot \theta\)

Step-by-Step Solution

Answer

\(\csc \theta - \sin \theta = \cos \theta \cdot \cot \theta\)

Step 1: Rewrite \(\csc \theta\) and \(\bout\theta\)

To begin, express \(\csc \theta\) and \(\cot \theta\) in terms of \(\sin \theta\) and \(\cos \theta\). Recall that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Step 2: Substitute and Simplify LHS

Substitute \(\csc \theta\) with \(\frac{1}{\sin \theta}\) on the left-hand side (LHS) of the equation: \(\frac{1}{\sin \theta} - \sin \theta\). To combine these terms, find a common denominator: \(\frac{1 - \sin^2 \theta}{\sin \theta}\).

Step 3: Use Pythagorean Identity

Remember the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Using this identity, replace \(1 - \sin^2 \theta\) with \(\cos^2 \theta\): \(\frac{\cos^2 \theta}{\sin \theta} \).

Step 4: Simplify the Expression

Simplify the fraction: \(\frac{\cos^2 \theta}{\sin \theta} = \cos \theta \cdot \frac{\cos \theta}{\sin \theta}\). Since \(\frac{\cos \theta}{\sin \theta} = \cot \theta\), the expression becomes \(\cos \theta \cdot \cot \theta\).

Step 5: Verify Both Sides

We've shown that the left-hand side \(\csc \theta - \sin \theta\) simplifies to \(\cos \theta \cdot \cot \theta\), which matches the right-hand side. Thus, the identity is established.

Key Concepts

Cosecant Function

The cosecant function, denoted as \(\csc \theta\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. In mathematical terms:
\[ \csc \theta = \frac{1}{\sin \theta} \]
This means for any angle \(\theta\), the cosecant is found by taking the inverse of the sine value. For example, if \(\sin \theta = 0.5\), then \(\csc \theta = 2.0\). The cosecant function is particularly useful in various calculus and geometry problems.
Remember, since \(\sin \theta\) is in the denominator, \(\csc \theta\) is undefined whenever \(\theta\) is an integer multiple of \(\pi\).

Cotangent Function

The cotangent function, denoted as \(\cot \theta\), is another of the six primary trigonometric functions. It is defined as the reciprocal of the tangent function. In terms of sine and cosine:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
This means cotangent relates cosine and sine in a single ratio. For example, if \(\cos \theta = 0.866\) and \(\sin \theta = 0.5\), then \(\cot \theta = \frac{0.866}{0.5} = 1.732\). The cotangent function is useful in problems involving angles and slopes and is essential in calculus for integrals involving trigonometric functions.
Cotangent, like cosecant, is undefined when \(\sin \theta\) is zero, corresponding to integer multiples of \(\pi\).

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that originates from the Pythagorean theorem. It states:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This identity is crucial as it connects the sine and cosine functions and is used extensively to simplify trigonometric expressions. For instance, if you know \(\sin^2 \theta\), you can find \(1 - \sin^2 \theta\) and replace it with \(\cos^2 \theta\). This was precisely done in the provided exercise to transform the expression \(1 - \sin^2 \theta\) into \(\cos^2 \theta\). The Pythagorean identity is also the basis for deriving other trigonometric identities involving tangent and secant.

Simplifying Expressions

Simplifying trigonometric expressions often involves rewriting functions in terms of sine and cosine, using identities, and performing algebraic manipulations. In the exercise, the original expression \(\csc \theta - \sin \theta\) was simplified step-by-step:

• Firstly, both \(\csc \theta\) and \(\cot \theta\) were rewritten in terms of \(\sin \theta\) and \(\cos \theta\).
• Then, a common denominator was found to combine terms into a single fraction.
• Next, the Pythagorean identity was applied to simplify the numerator.
• Lastly, the fraction was broken down, verifying that both sides of the equation matched.

Breaking expressions down into simpler components and step-by-step verification is the key to mastering trigonometric identities. Through practice, the process becomes more intuitive, streamlining the solution path significantly.