Problem 146 - Calculus Volume 1

Question

Simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. \(\frac{1+\tan ^{2} \alpha}{1+\cot ^{2} \alpha}\)

Step-by-Step Solution

Answer

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

Step 1: Convert Tangent and Cotangent to Sine and Cosine

Start by expressing \(\tan \alpha\) and \(\cot \alpha\) in terms of sine and cosine. Recall that \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\) and \(\cot \alpha = \frac{\cos \alpha}{\sin \alpha}\).

Step 2: Write the Given Expression in Sines and Cosines

Plug the expressions for \(\tan \alpha\) and \(\cot \alpha\) into the original expression: \[ \frac{1+\tan^2 \alpha}{1+\cot^2 \alpha} = \frac{1+\left(\frac{\sin \alpha}{\cos \alpha}\right)^2}{1+\left(\frac{\cos \alpha}{\sin \alpha}\right)^2} \] Simplify further as: \[ \frac{1+\frac{\sin^2 \alpha}{\cos^2 \alpha}}{1+\frac{\cos^2 \alpha}{\sin^2 \alpha}} \] which becomes: \[ \frac{\frac{\cos^2 \alpha + \sin^2 \alpha}{\cos^2 \alpha}}{\frac{\sin^2 \alpha + \cos^2 \alpha}{\sin^2 \alpha}} \]

Step 3: Simplify Using the Pythagorean Identity

Utilize the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). This transforms the expression further: \[ \frac{\frac{1}{\cos^2 \alpha}}{\frac{1}{\sin^2 \alpha}} \]

Step 4: Evaluate the Simplified Expression

To simplify, apply the division of fractions, which involves multiplying by the reciprocal. So:\[ \frac{\sin^2 \alpha}{\cos^2 \alpha} \]which equals \(\tan^2 \alpha\), arriving at the simplest form of the expression.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relation in trigonometry connecting the squares of sine and cosine functions. It states that for any angle \( \alpha \), \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This identity is based on the geometric interpretation of right-angled triangles and the Pythagorean Theorem.
\[ \text{(Hypotenuse)}^2 = \text{(Opposite Side)}^2 + \text{(Adjacent Side)}^2 \]
This forms the basis for the trigonometric identity because, in a unit circle, the radius is 1. The sin and cos of any angle are the lengths of the opposite and adjacent sides, respectively. Therefore, for any angle on a unit circle:

• The square of the sine of the angle plus the square of the cosine of the angle is 1.

When you see expressions like \( \sin^2 \alpha + \cos^2 \alpha \), this identity allows you to simplify them to 1, making subsequent calculations easier and often revealing elegant simplifications.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. They are critical tools for simplifying expressions and solving equations in trigonometry.
Some of the most commonly used identities are:

• The Pythagorean Identities: \( \sin^2 \alpha + \cos^2 \alpha = 1 \)
• Reciprocal Identities: \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \)
• Even-Odd Identities: These identities indicate how the sign of the functions change with negative angles like \( \sin(-\alpha) = -\sin \alpha \) and \( \cos(-\alpha) = \cos \alpha \)

Using these identities, you can transform and simplify trigonometric expressions by expressing them in terms of basic sine and cosine functions. This is a powerful step in solving more complex problems, as shown when converting \( \tan \alpha \) and \( \cot \alpha \) into their sine and cosine forms.

Tangent and Cotangent Relations

Tangent and cotangent functions are trigonometric ratios related to sine and cosine. Understanding their relationships helps in simplifying expressions involving these functions. The tangent \( \tan \alpha \) is defined as the ratio of the sine to the cosine of angle \( \alpha \):
\[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \]
Similarly, the cotangent \( \cot \alpha \) is the reciprocal of the tangent:
\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \]
These definitions enable the simplification of complex expressions such as \( \frac{1 + \tan^2 \alpha}{1 + \cot^2 \alpha} \).

• Express each tangent and cotangent in terms of sine and cosine.
• Use those expressions to substitute back, simplifying further using their relationships and trigonometric identities.

In this way, expressions that initially seem complex can be rewritten in simpler forms, seeing, in this case, that everything reduces neatly to \( \tan^2 \alpha \).