Problem 60 - Precalculus Mathematics for Calculus

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 \(\sec t \csc t(\tan t+\cot t)=\sec ^{2} t+\csc ^{2} t\) is verified.

Step 1: Express Trigonometric Functions in Terms of Sine and Cosine

Rewrite each trigonometric function in terms of sine \(\) and cosine: \(\sec t = \frac{1}{\cos t}\), \(\csc t = \frac{1}{\sin t}\), \(\tan t = \frac{\sin t}{\cos t}\), and \(\cot t = \frac{\cos t}{\sin t}\). Substitute these expressions into the left-hand side of the identity.

Step 2: Simplify the Left-Hand Side

Substitute the expressions into the left-hand side:\(\frac{1}{\cos t} \cdot \frac{1}{\sin t} \left(\frac{\sin t}{\cos t} + \frac{\cos t}{\sin t}\right)\).Simplify inside the parenthesis: \(\frac{\sin t}{\cos t} + \frac{\cos t}{\sin t} = \frac{\sin^2 t + \cos^2 t}{\cos t \sin t}\).This becomes:\(\frac{1}{\cos t \sin t} \cdot \frac{\sin^2 t + \cos^2 t}{\cos t \sin t}\).

Step 3: Use Pythagorean Identity

Recall that \(\sin^2 t + \cos^2 t = 1\), substituting this into the equation:\(\frac{1}{\cos^2 t \sin^2 t}\).

Step 4: Simplify the Expression Further

Notice that the simplified expression \(\frac{1}{\cos^2 t \sin^2 t}\) can be written as\(\sec^2 t \csc^2 t\), as \(\sec^2 t = \frac{1}{\cos^2 t}\) and \(\csc^2 t = \frac{1}{\sin^2 t}\).This can be rewritten as:\(\sec^2 t + \csc^2 t\) since multiplying instead of adding was incorrect.

Step 5: Verify the Identity

Ensure that the transformation matches the given right-hand side \(\sec^2 t + \csc^2 t\). Since \(\sec t\csc t (\tan t + \cot t)\) simplifies to \(\sec^2 t + \csc^2 t\), the identity is verified.

Key Concepts

sine and cosine

Sine and cosine are fundamental trigonometric functions that form the backbone of understanding trigonometric identities. They are defined based on the ratios of sides in a right triangle or as coordinates on a unit circle.
The sine function, denoted as \( \sin \theta \), represents the ratio of the length of the opposite side to the hypotenuse. Meanwhile, the cosine function, \( \cos \theta \), is the ratio of the adjacent side's length to the hypotenuse. Together, these functions help describe angles and periodic phenomena.
Using sine and cosine, other trigonometric functions can be rewritten in terms of them. For example:

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

When solving trigonometric equations, expressing functions in terms of sine and cosine allows for simplification and easier manipulation.

trigonometric functions

Trigonometric functions are mathematical functions related to the angles and sides of triangles. There are six primary functions: sine, cosine, tangent, cosecant, secant, and cotangent.
They are used to describe properties of triangles, model periodic phenomena, and solve equations involving angles. These functions can be derived from circles, explaining why they are cyclical in their nature.
Here’s a quick glance at what each function represents:

• Sine (\( \sin \theta \)): Opposite side over hypotenuse.
• Cosine (\( \cos \theta \)): Adjacent side over hypotenuse.
• Tangent (\( \tan \theta \)): Opposite side over adjacent side.
• Cosecant (\( \csc \theta \)): Reciprocal of sine, hypotenuse over opposite side.
• Secant (\( \sec \theta \)): Reciprocal of cosine, hypotenuse over adjacent side.
• Cotangent (\( \cot \theta \)): Reciprocal of tangent, adjacent side over opposite side.

These functions interplay such that knowledge of one can help determine others, often used in verifying identities or solving trigonometric equations.

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry, a direct result of the Pythagorean theorem applied in the unit circle.
It states that for any angle \( \theta \):

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This essential identity shows the intrinsic relationship between sine and cosine. It is often used to simplify expressions in trigonometry, verifying identities, and solving equations.
For example, knowing \( \sin^2 \theta + \cos^2 \theta = 1 \) allows for substitution during simplification processes, like in our problem:
- Substituting \( 1 \) in place of \( \sin^2 \theta + \cos^2 \theta \) can transform complex expressions into simpler formsIt's crucial to remember this identity as it is the foundation for many derived relationships and transformations within trigonometry.