Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\tan \theta \cot \theta=1$$

Step-by-Step Solution

Answer

The identity \( \tan \theta \cot \theta = 1 \) is verified as true.

Step 1: Express Tangent and Cotangent in Terms of Sine and Cosine

Start by expressing both \( \tan \theta \) and \( \cot \theta \) using sine and cosine functions. We have:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]

Step 2: Multiply the Two Expressions

Multiply the initial expressions of \( \tan \theta \) and \( \cot \theta \) derived in Step 1:\[ \tan \theta \cdot \cot \theta = \left( \frac{\sin \theta}{\cos \theta} \right) \cdot \left( \frac{\cos \theta}{\sin \theta} \right) \]

Step 3: Simplify the Expression

Notice that when you multiply the two fractions, the numerator of the first fraction \( \sin \theta \) and the denominator of the second fraction \( \sin \theta \) cancel each other out, as do the \( \cos \theta \) terms:\[ \frac{\sin \theta \cdot \cos \theta}{\cos \theta \cdot \sin \theta} = \frac{\sin \theta}{\sin \theta} \cdot \frac{\cos \theta}{\cos \theta} = 1 \]

Step 4: Conclusion

After simplifying, we have shown that the original expression \( \tan \theta \cdot \cot \theta \) equals 1. Therefore, the identity is verified to be true, proving \( \tan \theta \cot \theta = 1 \).

Key Concepts

Tangent

The tangent of an angle in a right triangle is an important trigonometric function. It is defined as the ratio of the length opposite the angle to the length of the adjacent side. In mathematical terms, tangent can be represented as:

• \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)

However, in terms of the unit circle and sine and cosine functions, tangent can be expressed as follows:

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

This representation is particularly useful when proving identities, as it allows us to transform expressions into a form that may be simpler to work with. You can see how the tangent function is used to simplify expressions and verify identities, like the one discussed where \( \tan \theta \cdot \cot \theta = 1 \). This expression uses the property that when sine and cosine are transformed, their respective products simplify to 1.

Cotangent

Cotangent is another fundamental trigonometric function, often used to provide an alternative perspective on relationships between angles and sides. It is defined as the reciprocal of the tangent of an angle. Mathematically, this is expressed as:

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

The cotangent function can be viewed as flipping the ratio of tangent. Given it is a reciprocal, it offers a unique identity when combined with tangent. As shown in the exercise, multiplying the tangent and cotangent of the same angle results in 1:

• \(\tan \theta \cdot \cot \theta = 1\)

This identity is valuable in proofs and simplifying complex trigonometric expressions, especially because it combines the fundamental sine and cosine relationships in such a straightforward equation.

Sine and Cosine Transformations

Sine and cosine functions are the building blocks of trigonometry. They are foundational in defining other trigonometric functions like tangent and cotangent. These functions can be used to transform expressions into forms that are simpler to analyze and manipulate.With the unit circle, sine and cosine have simple definitions:

• Sine (\( \sin \theta \)): Length of the vertical side of the right triangle where the hypotenuse is the radius.
• Cosine (\( \cos \theta \)): Length of the horizontal side of the right triangle where the hypotenuse is the radius.

Transformations involving sine and cosine include the conversion of tangent and cotangent into their respective sine and cosine ratios. These transformations greatly simplify the process of verifying identities or solving equations. For example, converting \( \tan \theta \) into \( \frac{\sin \theta}{\cos \theta} \) allows us to see a straightforward multiplication and division between sine and cosine leading to expressions like \( \tan \theta \cdot \cot \theta = 1 \). This is a powerful demonstration of how transformations help in checking and proving trigonometric identities effortlessly.