Problem 20 - Algebra and Trigonometry

Question

\(19-22\) . Prove the cofunction identity using the addition and subtraction formulas. $$ \cot \left(\frac{\pi}{2}-u\right)=\tan u $$

Step-by-Step Solution

Answer

The identity \(\cot(\frac{\pi}{2}-u) = \tan u\) is proven using the cotangent subtraction formula.

Step 1: Understanding the Identity

We need to prove the cofunction identity for \ \(\cot \left(\frac{\pi}{2}-u\right)=\tan u\) using the addition and subtraction formulas for trigonometric functions.

Step 2: Applying Cotangent Subtraction Identity

Recall that the cotangent of a difference is given by: \ \[ \cot(A-B) = \frac{\cot A \cot B + 1}{\cot B - \cot A} \] \ Applying this to \(A = \frac{\pi}{2}\) and \(B = u\), we have \ \[ \cot\left(\frac{\pi}{2} - u\right) = \frac{\cot\frac{\pi}{2} \cot u + 1}{\cot u - \cot\frac{\pi}{2}} \] Since \( \cot\frac{\pi}{2} = 0 \), the formula simplifies to:

Step 3: Simplification of the Cotangent Expression

Substitute \(\cot\frac{\pi}{2} = 0\) into the expression: \[ \cot\left(\frac{\pi}{2} - u\right) = \frac{0 \cdot \cot u + 1}{\cot u - 0} \]This further simplifies to:\[ \cot\left(\frac{\pi}{2} - u\right) = \frac{1}{\cot u} \] Since \(\cot u = \frac{1}{\tan u}\), we have \(\frac{1}{\cot u} = \tan u\).

Step 4: Conclusion

Therefore, by substitution, we confirm that \( \cot\left(\frac{\pi}{2} - u\right) = \tan u \), proving the original cofunction identity using the subtraction formula for trigonometric functions.

Key Concepts

Tan Function

The tangent function, often represented as \( \tan \theta \), is one of the basic trigonometric functions. In the context of a right triangle, it is defined as the ratio of the opposite side to the adjacent side of the angle \( \theta \). Visually, this is expressed as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

Beyond right triangles, the tangent function is also defined on the unit circle. It is the slope of the line that passes through the origin and the point corresponding to the angle on the unit circle.
The function has a periodicity of \( \pi \) radians, repeating every 180 degrees. It has vertical asymptotes at \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer, as the function becomes undefined at these points.
Understanding the tangent function is crucial because it frequently appears in trigonometric identities and equations, making knowledge of its behavior and properties essential in solving trigonometric problems.

Cot Function

The cotangent function, denoted as \( \cot \theta \), is the reciprocal of the tangent function. In a right triangle, it is defined as the ratio of the adjacent side to the opposite side:

• \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{\tan \theta} \)

The cotangent function, like tangent, can also be derived from the unit circle. It is defined as the reciprocation of the tangent at each angle \( \theta \) around the circle. It shares a periodicity of \( \pi \) radians, similar to the tangent function.
The cotangent is undefined where the tangent is zero, leading to vertical asymptotes at these points. This typically occurs at \( \theta = k\pi \), where \( k \) is an integer. In trigonometric identities, the cotangent often appears in relationships involving reciprocal or cofunction identities. Grasping its properties is vital to understanding and solving trigonometric equations.

Addition and Subtraction Formulas

Addition and subtraction formulas are powerful tools in trigonometry. They allow us to find the values of trigonometric functions for sums or differences of angles. For instance, the cotangent subtraction formula is:\[ \cot(A - B) = \frac{\cot A \cot B + 1}{\cot B - \cot A} \]These formulas enable us to break down complex trigonometric expressions into simpler forms, facilitating easier manipulation and problem-solving.
Using these, we can derive other identities and simplify expressions involving multiple angles. Addition and subtraction formulas become especially useful when working with angles that do not have standard known values. Their application can be seen in transformations, analytical geometry, and various proofs in trigonometry, such as proving cofunction identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are foundational in trigonometry, providing relationships between different trigonometric functions. Some fundamental identities include:

• Pythagorean Identities: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Reciprocal Identities: \( \tan \theta = \frac{1}{\cot \theta} \)
• Cofunction Identities: \( \sin(\frac{\pi}{2} - \theta) = \cos \theta \)

Trigonometric identities simplify solving trigonometric equations and proving expressions in calculus and algebra. They can transform complex expressions, leading to the simplification needed for easier calculation or deeper insight into a function's behavior. Mastery of these identities forms a critical skill for anyone looking to excel in mathematics and related fields.