Problem 58 - Algebra and Trigonometry

Question

Verify the identity. $$ \sec t \csc t(\tan t+\cot t)=\sec ^{2} t+\csc ^{2} t $$

Step-by-Step Solution

Answer

The trigonometric identity is verified: the expressions are equal.

Step 1: Expand the Left Side

First, expand the left side of the equation \( \sec t \csc t(\tan t + \cot t) \). This means multiplying each term in the parentheses by \( \sec t \csc t \). Begin with:\[\sec t \csc t \cdot \tan t + \sec t \csc t \cdot \cot t\]

Step 2: Rewrite Using Trigonometric Identities

Rewrite the expressions using fundamental trigonometric identities. We know:- \( \tan t = \frac{\sin t}{\cos t} \)- \( \cot t = \frac{\cos t}{\sin t} \)- \( \sec t = \frac{1}{\cos t} \)- \( \csc t = \frac{1}{\sin t} \)Now substitute these into the expanded form:\[\frac{1}{\cos t} \cdot \frac{1}{\sin t} \cdot \frac{\sin t}{\cos t} + \frac{1}{\cos t} \cdot \frac{1}{\sin t} \cdot \frac{\cos t}{\sin t}\]

Step 3: Simplify Each Term

Simplify each fraction from Step 2:- The first term: \( \frac{1}{\cos t} \cdot \frac{1}{\sin t} \cdot \frac{\sin t}{\cos t} = \frac{1}{\cos^2 t} = \sec^2 t \)- The second term: \( \frac{1}{\cos t} \cdot \frac{1}{\sin t} \cdot \frac{\cos t}{\sin t} = \frac{1}{\sin^2 t} = \csc^2 t \)Add these results:\[\sec^2 t + \csc^2 t\]

Step 4: Compare to the Right Side

Now compare the simplified left side \( \sec^2 t + \csc^2 t \) with the right side of the equation. They are identical, confirming the identity. Thus:\[\sec t \csc t (\tan t + \cot t) = \sec^2 t + \csc^2 t\] The identity is verified.

Key Concepts

Secant Identity

The secant function, denoted as \( \sec t \), is one of the fundamental trigonometric functions. It is the reciprocal of the cosine. This means

• \( \sec t = \frac{1}{\cos t} \)

Understanding this relationship is crucial when working with trigonometric identities, as it allows for straightforward transformations and simplifications.

When verifying trigonometric identities, converting functions to their cosine or sine equivalents often simplifies the process. For example, using the secant identity can reveal patterns and make complex trigonometric equations more manageable. In the context of the original exercise, employing \( \sec t = \frac{1}{\cos t} \) enabled the simplification of terms on the left side of the equation. Such transformations are essential tools in trigonometry.

Cosecant Identity

The cosecant function, denoted by \( \csc t \), is the reciprocal of the sine function. Therefore, it is defined as:

• \( \csc t = \frac{1}{\sin t} \)

Similar to the secant identity, using the cosecant identity provides a pathway to simplify trigonometric expressions.

In the original exercise, transforming expressions with \( \csc t \) into terms involving \( \sin t \) helped in achieving the simplification necessary for verifying the identity. By expressing trigonometric functions in terms of sine and cosine, complex equations become simpler, often reducing to basic identities involving simple arithmetic or algebraic manipulation. This approach highlights the strategic use of identities to streamline solving trigonometric problems.

Tangent and Cotangent Identities

Tangent, denoted as \( \tan t \), and its reciprocal, cotangent \( \cot t \), also serve vital roles in trigonometric simplifications. They are defined as:

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

These identities allow us to express tangent and cotangent in terms of sine and cosine, which are easier to work with when verifying identities.

In the exercise, using the tangent and cotangent identities was critical for checking the trigonometric equation. Once the expressions were rewritten using sine and cosine, it became possible to identify and cancel components, simplifying the given expression to match the right side of the equation. This strategic substitution facilitated the verification process, which demonstrates how these identities can simplify complex equations in trigonometry.