Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\frac{1+\tan ^{2} \theta}{1+\cot ^{2} \theta}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \tan^2 \theta \).

Step 1: Rewrite tangent and cotangent

To start, rewrite the tangent and cotangent functions using their expressions in terms of sine and cosine. Recall:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \] Therefore, \( \tan^2 \theta = \left( \frac{\sin \theta}{\cos \theta} \right)^2 \) and \( \cot^2 \theta = \left( \frac{\cos \theta}{\sin \theta} \right)^2 \).

Step 2: Substitute into the expression

Substitute these identities into the given expression:\[ \frac{1 + \left(\frac{\sin \theta}{\cos \theta}\right)^2}{1 + \left(\frac{\cos \theta}{\sin \theta}\right)^2} \] This simplifies to:\[ \frac{1 + \frac{\sin^2 \theta}{\cos^2 \theta}}{1 + \frac{\cos^2 \theta}{\sin^2 \theta}} \]

Step 3: Simplify each part of the fraction

Simplify each component:The numerator becomes:\[ 1 + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta} \]Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), this becomes:\[ \frac{1}{\cos^2 \theta} = \sec^2 \theta \]The denominator becomes:\[ 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} \]Using the same identity, this becomes:\[ \frac{1}{\sin^2 \theta} = \csc^2 \theta \]

Step 4: Final expression

Combine these results to find the simplified expression:\[\frac{\sec^2 \theta}{\csc^2 \theta} = \frac{\frac{1}{\cos^2 \theta}}{\frac{1}{\sin^2 \theta}} = \frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta\]

Step 5: Conclusion

The expression \( \frac{1+\tan^2 \theta}{1+\cot^2 \theta} \) simplifies to \( \tan^2 \theta \).

Key Concepts

Sine and Cosine

Sine and cosine are foundational trigonometric functions that enable us to express relationships in triangles and circles. They connect angles to side lengths in right-angled triangles, and are defined on the unit circle. If we imagine a point making an angle \( \theta \) with the positive x-axis, the coordinates of this point are \( (\cos \theta, \sin \theta) \).
The sine function, represented by \( \sin \theta \), gives the y-coordinate (vertical position) of the point. Meanwhile, the cosine function, \( \cos \theta \), provides the x-coordinate (horizontal position).
These functions create a basis for many trigonometric identities. A key relationship involving sine and cosine is the Pythagorean identity: \( \cos^2 \theta + \sin^2 \theta = 1 \). This identity is crucial for simplifying expressions because it can replace complex terms with simpler equivalents. Understanding how to manipulate sine and cosine is essential for solving trigonometric problems effectively.

Tangent and Cotangent

Tangent (\( \tan \theta \)) and cotangent (\( \cot \theta \)) are functions derived from sine and cosine. The tangent is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This means tangent provides the ratio of the opposite side to the adjacent side in a right-angled triangle.
The cotangent, the reciprocal of tangent, is \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This gives the ratio of the adjacent side to the opposite side. These two functions are particularly useful when working with angles or slopes.
When you square these functions, \( \tan^2 \theta = \left( \frac{\sin \theta}{\cos \theta} \right)^2 \) and \( \cot^2 \theta = \left( \frac{\cos \theta}{\sin \theta} \right)^2 \), their relationships in terms of sine and cosine allow easier manipulation of expressions. Understanding these interactions provides powerful tools to solve problems involving trigonometric expressions and equations.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex trigonometric forms into simpler components. This can often mean expressing functions like tangent and cotangent in terms of basic trigonometric identities like sine and cosine.
For example, in the given problem, an understanding of trigonometric identities allows us to express \( \tan^2 \theta \) and \( \cot^2 \theta \) in terms of sine and cosine. The Pythagorean identity is then used to further simplify the expressions.
Steps to simplify include substituting trigonometric identities, using mathematical laws such as reciprocal identities, and reducing fractions. Simplified expressions often lead to discovering new forms or relationships, making complex calculations more manageable. A broad understanding of these strategies is fundamental for mastering trigonometry concepts.