Problem 52 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$(\tan y+\cot y) \sin y \cos y=1$$

Step-by-Step Solution

Answer

The identity \((\tan y + \cot y) \sin y \cos y = 1\) is verified as true using trigonometric identities.

Step 1: Expand the Expression

Start by writing down the given expression: \((\tan y + \cot y) \sin y \cos y\). We need to simplify this expression. Start by expressing \(\tan y\) and \(\cot y\) in terms of \(\sin y\) and \(\cos y\): \(\tan y = \frac{\sin y}{\cos y}\) and \(\cot y = \frac{\cos y}{\sin y}\).

Step 2: Substitute Trigonometric Functions

Replace \(\tan y\) and \(\cot y\) in the equation: \((\frac{\sin y}{\cos y} + \frac{\cos y}{\sin y}) \sin y \cos y\).

Step 3: Simplify the Expression

Combine the terms: \(\frac{\sin^2 y}{\cos y} + \frac{\cos^2 y}{\sin y}\). Now we can multiply each term by \(\sin y \cos y\): \(\sin^2 y + \cos^2 y\).

Step 4: Use the Pythagorean Identity

Recall the Pythagorean identity: \(\sin^2 y + \cos^2 y = 1\). According to this identity, the expression simplifies to \(1\).

Step 5: Conclusion

Thus, \((\tan y + \cot y) \sin y \cos y = 1\) is verified, completing the proof.

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics that relate the angles of triangles to the lengths of their sides. They are especially useful in various fields such as physics, engineering, and astronomy.

In the context of the given problem, we worked with two specific trigonometric functions: tangent (\( an y\)) and cotangent (\( ext{cot} y\)). Here's what they represent:

• **Tangent (\(\tan y\))**: Defined as the ratio of the opposite side to the adjacent side in a right triangle. In terms of sine and cosine, it is expressed as \(\frac{\sin y}{\cos y}\).
• **Cotangent (\(\cot y\))**: The reciprocal of tangent, defined as the ratio of the adjacent side to the opposite side. In terms of sine and cosine, it is expressed as \(\frac{\cos y}{\sin y}\).

We often use these definitions to transform and simplify trigonometric expressions in equations, as shown in the solution of this exercise.

Simplifying Expressions

Simplifying expressions in trigonometry often requires substituting certain functions with their equivalent forms. This transformation makes it easier to apply mathematical identities and proves theorems or verify equalities.

In our example, by substituting \(\tan y\) with \(\frac{\sin y}{\cos y}\) and \(\cot y\) with \(\frac{\cos y}{\sin y}\), we were able to rewrite the original expression. This substitution allowed for further simplification by combining and multiplying terms effectively.

Simplifying the expression in such a way reduces complexity and helps clearly see the relationship between different elements of the expression. Ultimately, it brings us closer to confirming the given identity or equation.

Pythagorean Identity

The Pythagorean identity is one of the core concepts in trigonometry. It stems from the Pythagorean theorem in geometry, which deals with right triangles. The identity states that for any angle \(y\):
\[\sin^2 y + \cos^2 y = 1\]

This identity is incredibly useful in simplifying expressions and proving various trigonometric identities, as it shows the intrinsic relationship between sine and cosine functions.

In our specific exercise, after transforming and simplifying the expression, we reached \(\sin^2 y + \cos^2 y\). By applying the Pythagorean identity, we immediately concluded that the expression simplifies to 1. This step was crucial in verifying the original identity provided in the problem, showcasing how fundamental such identities are in solving trigonometric problems.