Problem 32 - Precalculus Essentials

Question

In Exercises \(23-34\), verify each identity. $$\sin 2 t-\cot t=-\cot t \cos 2 t$$

Step-by-Step Solution

Answer

The identity \(\sin 2 t-\cot t=-\cot t \cos 2 t\) holds true for all real numbers \(t\), where \(\tan t\) or \(\sin t\) is not equal to zero.

Step 1: Applying Trigonometric Identities

Firstly, replace \(\sin 2t\) by \(2\sin t \cos t\) and \(\cos 2t\) by \(1-2\sin^2t\) on the right side and simplify, such that the equation becomes: \[2\sin t \cos t-\cot t = -\cot t(1-2\sin^2t)\].

Step 2: Substitute Cotangent by Reciprocal of Tangent

The cotangent function can be expressed as the reciprocal of tangent. Substitute \(\cot t\) with \(\frac{1}{\tan t}\) and simplify the equation again: \[2\sin t \cos t-\frac{1}{\tan t} = -\frac{1}{\tan t}(1-2\sin^2t)\].

Step 3: Expressing in terms of Sine and Cosine

Express tangent in terms of sine and cosine as well to get: \[2\sin t \cos t-\frac{\cos t}{\sin t} = -\frac{\cos t}{\sin t}(1-2\sin^2t)\] Now, convert this tabular form to allow for easier simplification and comparison.

Step 4: Simplifying and comparing both sides

On simplifying and reordering terms, both sides of the equation turn into \(\cos t(\sin t-\frac{1}{\sin t})\), confirming that the initial trigonometric equation was correct.

Key Concepts

Understanding Sine and Cosine

Sine and Cosine are fundamental trigonometric functions, expressed as \( \sin \theta \) and \( \cos \theta \) respectively. They are based on the angles of a right triangle, defined as:

• \( \sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}} \)
• \( \cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)

These two functions help in simplifying many trigonometric expressions, especially when identities come into play. A key identity to remember is the double-angle identity. For sine, \( \sin 2t = 2 \sin t \cos t \). For cosine, \( \cos 2t = 1 - 2\sin^2 t \) or \( \cos^2 t - \sin^2 t \).
These identities allow for transformation and simplification of trigonometric expressions into more manageable forms, as seen in the step-by-step solution where replacements simplify the problem.

Exploring Cotangent and Tangent

Cotangent and tangent are reciprocals of each other. The tangent of an angle \( t \), written as \( \tan t \) is given by:

• The ratio of sine to cosine: \( \tan t = \frac{\sin t}{\cos t} \).

On the flip side, cotangent, written as \( \cot t \) is:

• The reciprocal of tangent: \( \cot t = \frac{\cos t}{\sin t} \).

By substituting cotangent with this reciprocal identity, challenging trigonometric expressions become more approachable. In the exercise, this substitution is used in simplifying the equation, as cotangent is replaced with its expression involving sine and cosine. Simplifying such expressions often requires recognizing these relationships and converting one form to another.

Techniques in Trigonometric Simplification

Trigonometric simplification is all about transforming complex expressions into simpler ones using identities. This involves a few strategic steps:

• Substitute known identities (like double-angle identities).
• Convert all trigonometric functions to sine and cosine for uniformity.
• Simplify by finding common terms and factoring.

In the exercise, simplification began by expressing the compound functions, like \( \sin 2t \) and \( \cos 2t \), in simpler terms and then substituting the cotangent with a sine-cosine expression. This method allows for factoring common elements and thus confirming both sides of the identity match. Remember, simplified expressions are often much easier to evaluate or compare, ensuring the correctness of mathematical identities.