Problem 60 - 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 identity is verified: \( \sec t \csc t (\tan t + \cot t) = \sec^2 t + \csc^2 t \).

Step 1: Simplify the Left Side of the Equation

We start by expanding the left side of the equation, \( \sec t \csc t (\tan t + \cot t) \). Distribute \( \sec t \csc t \) across both terms inside the parentheses to get:\[ \sec t \csc t \tan t + \sec t \csc t \cot t. \]

Step 2: Use Trigonometric Identities

Recall the trigonometric identities for \( \tan t = \frac{\sin t}{\cos t} \), \( \cot t = \frac{\cos t}{\sin t} \), \( \sec t = \frac{1}{\cos t} \), and \( \csc t = \frac{1}{\sin t} \). Substitute these into the expression:\[ \left( \frac{1}{\cos t} \right) \left( \frac{1}{\sin t} \right) \left( \frac{\sin t}{\cos t} \right) + \left( \frac{1}{\cos t} \right) \left( \frac{1}{\sin t} \right) \left( \frac{\cos t}{\sin t} \right). \]

Step 3: Simplify Each Term

Simplify the first term:\[ \frac{1}{\cos t} \cdot \frac{1}{\sin t} \cdot \frac{\sin t}{\cos t} = \frac{1 \cdot \sin t}{\cos^2 t \cdot \sin t} = \frac{1}{\cos^2 t} = \sec^2 t. \]Simplify the second term:\[ \frac{1}{\cos t} \cdot \frac{1}{\sin t} \cdot \frac{\cos t}{\sin t} = \frac{1 \cdot \cos t}{\sin^2 t \cdot \cos t} = \frac{1}{\sin^2 t} = \csc^2 t. \]

Step 4: Combine the Simplified Terms

Add the simplified results of the previous step:\[ \sec^2 t + \csc^2 t. \]This matches the right-hand side of the original equation, confirming that the identity:\( \sec t \csc t (\tan t + \cot t) = \sec^2 t + \csc^2 t \) is valid.

Key Concepts

Secant

The secant function is one of the six essential trigonometric functions, though not as commonly used as sine and cosine. Mathematically, the secant of an angle, often denoted as \( heta \), is the reciprocal of the cosine:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This definition is key because it shows that secant is undefined when the cosine of the angle is zero, meaning it's important to remember angle restrictions when dealing with secant.
Secant plays an important role in trigonometry when solving problems that involve division by cosine or need to work with its reciprocal. It's very useful in simplifying expressions involving complex trigonometric identities and verifying equations like you've seen in your original exercise.
Additionally, secant is closely connected with the Pythagorean identity \(1 + \tan^2 \theta = \sec^2 \theta\), which further aids in manipulating and understanding the relationship between different trigonometric entities.

Cosecant

Similar to secant, the cosecant function is another reciprocal trigonometric function. It's the reciprocal of sine and is denoted as \(\csc\theta\). Mathematically, it is expressed as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Since it is the reciprocal, cosecant becomes undefined when sine is zero. This aspect means while working with \( \csc \theta \), it’s crucial to consider these undefined points to avoid incorrect calculations.
Cosecant is useful particularly in problem-solving scenarios that involve ratios of triangles, especially when sine components are involved directly or reciprocally.
The identity \( 1 + \cot^2 \theta = \csc^2 \theta \) is quite helpful and can also simplify equations much like the original problem given to you, which involved ties with \(\csc^2 t\). Understanding these relationships and identities can provide a far easier pathway to proving or solving trigonometric identities.

Tangent

Tangent is among the primary trigonometric functions, and it often appears in various mathematical contexts. Defined geometrically, tangent relates the sides of a right triangle, giving the ratio of the opposite side to the adjacent side.

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This ratio becomes undefined where \(\cos \theta\) equals zero. Thus, care is needed in equations to recognize where tangent could potentially introduce limits and discontinuities.
Tangent is well-integral to the identity \( \sec^2 \theta = 1 + \tan^2 \theta \), which is part of what helps verify identities like the one in your textbook solution.
Utilizing tangent in these identities not only aids in understanding angles but also in offering a comprehensive view of how other trigonometric functions behave and relate in a coordinated system.

Cotangent

The cotangent, like its counterpart, tangent, is vital in many trigonometric expressions and identities. It is the reciprocal of tangent and gives the ratio of the adjacent side to the opposite side in a right triangle:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

Given its connection to tangent, cotangent becomes undefined exactly at the values where \( \tan \theta \) is zero, creating unique points to watch when solving equations.
Cotangent simplifies trigonometric expressions that might initially seem daunting, providing a different angle into resolving identities such as \( \csc^2 \theta = 1 + \cot^2 \theta \). This identity is frequently used to prove various trigonometric equations or to verify other identities, like in your task.
Understanding the behaviors and applications of cotangent can significantly enhance your capabilities in dealing with complex trigonometric identities.