Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\cot (-t)+\tan (-t)}{\cot t}=-\sec ^{2} t $$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\cot (-t)+\tan (-t)}{\cot t}=-\sec ^{2} t \).

Step 1: Simplify the Left Side of the Identity

Start by recalling the identities of trigonometric functions: \( \cot(-t) = -\cot t \) and \( \tan(-t) = -\tan t \). Substitute these into the left-hand side of the equation:\[ \frac{\cot(-t) + \tan(-t)}{\cot t} = \frac{-\cot t + (-\tan t)}{\cot t} = \frac{-\cot t - \tan t}{\cot t} \]

Step 2: Express in Terms of Sine and Cosine

Convert \( \cot t \) and \( \tan t \) to sine and cosine:\[ \cot t = \frac{\cos t}{\sin t} \] \[ \tan t = \frac{\sin t}{\cos t} \]Substitute these into the expression:\[ \frac{-\frac{\cos t}{\sin t} - \frac{\sin t}{\cos t}}{\frac{\cos t}{\sin t}} \]

Step 3: Find a Common Denominator

Find a common denominator for the numerator:\[ -\frac{\cos t}{\sin t} - \frac{\sin t}{\cos t} = -\left( \frac{\cos^2 t}{\sin t \cos t} + \frac{\sin^2 t}{\sin t \cos t} \right) = -\frac{\cos^2 t + \sin^2 t}{\sin t \cos t} \] Remember that \( \cos^2 t + \sin^2 t = 1 \). Substitute:\[ -\frac{1}{\sin t \cos t} \]

Step 4: Simplify the Fraction

Simplify the entire expression:\[ = \frac{-\frac{1}{\sin t \cos t}}{\frac{\cos t}{\sin t}} = -\frac{1}{\sin t \cos t} \times \frac{\sin t}{\cos t} \]Cancel out similar terms:\[ = -\frac{1}{\cos^2 t} \]

Step 5: Recognize as \(-\sec^2 t\)

Recall that \( \sec t = \frac{1}{\cos t} \), therefore:\[ -\frac{1}{\cos^2 t} = -\sec^2 t \].This matches the right-hand side of the original identity, confirming the identity is verified.

Key Concepts

Cotangent and Tangent Identities

In trigonometry, understanding cotangent and tangent identities is important. These identities help simplify trigonometric expressions.
\[ \text{Cotangent of } -t: \quad \cot(-t) = -\cot t \]
\[ \text{Tangent of } -t: \quad \tan(-t) = -\tan t \]
Both functions have negative angles leading to negative values of their positive angle counterparts.
Using these identities, mathematical expressions involving negative angles become easier to manage.
In our exercise, we started by using these identities to simplify the expression:
\[ \frac{\cot(-t) + \tan(-t)}{\cot t} \]
This identity becomes simplified to:
\[ \frac{-\cot t - \tan t}{\cot t} \].
This was the first step towards verifying the given identity. Identifying and applying these fundamental identities correctly is crucial for working through complex trigonometric problems.

Sine and Cosine Relationships

To solve complex trigonometric identities, expressing functions in terms of sine and cosine is often useful.
Tangent and cotangent can be represented using sine and cosine in the following way:

• Cotangent: \( \cot t = \frac{\cos t}{\sin t} \)
• Tangent: \( \tan t = \frac{\sin t}{\cos t} \)

In our exercise, we performed substitution using these relationships which led us to a more manageable form:
\[ \frac{-\frac{\cos t}{\sin t} - \frac{\sin t}{\cos t}}{\frac{\cos t}{\sin t}} \]Finding a common denominator, \( \sin t \cos t \), allowed us to combine terms in the numerator.
Remembering the Pythagorean identity, where \( \cos^2 t + \sin^2 t = 1 \), further simplified the expression to:
\[ -\frac{1}{\sin t \cos t} \]Understanding these basic sine and cosine relationships helps uncover hidden simplifications in trigonometric identities.

Secant Function

The secant function is a trigonometric function closely related to the cosine function.
Mathematically, secant is defined as:

• \( \sec t = \frac{1}{\cos t} \)

In trigonometric identities, we often encounter expressions involving \( \sec^2 t \).
In the exercise, once we simplified to \(-\frac{1}{\cos^2 t}\), it was crucial to recognize this as \(-\sec^2 t\).
This recognition allows us to complete the verification process of the identity.
\( \sec^2 t \) often appears in various trigonometric identities and differential calculus.
Having a clear understanding of how \( \sec \) relates to \( \cos \) is essential in simplifying and verifying advanced trigonometric identities and expressions.