Problem 37 - Algebra and Trigonometry

Question

Verify the identity. $$ \tan \theta+\cot \theta=\sec \theta \csc \theta $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \frac{1}{\sin \theta \cos \theta} \).

Step 1: Express in Terms of Sine and Cosine

To simplify and verify the identity, we first express each trigonometric function in terms of sine and cosine. The tangent function is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The cotangent function is expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). The secant function is expressed as \( \sec \theta = \frac{1}{\cos \theta} \). The cosecant function is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Simplify Left Side of the Equation

Now, let's simplify the left side of the equation: \( \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \). To add these two fractions, we need a common denominator. The common denominator is \( \sin \theta \cos \theta \). Thus, \( \frac{\sin \theta}{\cos \theta} = \frac{\sin^2 \theta}{\sin \theta \cos \theta} \) and \( \frac{\cos \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta \cos \theta} \). Adding them gives: \( \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \).

Step 3: Use Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substituting this into our equation, we have: \( \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \).

Step 4: Simplify Right Side of the Equation

Now, simplify the right side of the original equation: \( \sec \theta \csc \theta = \frac{1}{\cos \theta} \times \frac{1}{\sin \theta} = \frac{1}{\sin \theta \cos \theta} \).

Step 5: Verification

Both simplified sides of the equation were reduced to \( \frac{1}{\sin \theta \cos \theta} \). Thus, the equation is verified to be an identity as both sides are equal.

Key Concepts

Tangent

The tangent of an angle in a right-angled triangle is a fundamental trigonometric function. It is defined as the ratio of the opposite side to the adjacent side when considering the angle in question. Mathematically, this is expressed as:

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

In the context of the unit circle, which is another useful way to understand trigonometric functions, the tangent is represented by the ratio of the sine of the angle to the cosine of the angle:

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

This definition becomes particularly handy when transforming identities or equations because it allows you to express tangent using the basic trigonometric functions sine and cosine, which are usually easier to manipulate in proofs and calculations.
Understanding tangent can help solve many types of problems such as verifying identities, finding angles, and determining exact values.

Cotangent

Cotangent is another trigonometric function that is closely related to tangent, but it is its reciprocal. In right triangle terms, cotangent is the ratio of the adjacent side to the opposite side:

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

Similarly, in terms of sine and cosine, cotangent can be expressed as:

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

This reciprocal relationship with tangent makes cotangent a valuable identity to use when simplifying complex trigonometric expressions.
In many situations, expressing a function in terms of cotangent can simplify the process of solving trigonometric identities and equations. Also, recognizing when to use cotangent helps in integrating or differentiating trigonometric functions.

Secant

Secant is another essential trigonometric function, primarily used in scenarios where ratios involving cosine are needed. Secant is defined as the reciprocal of the cosine of an angle:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This function arises naturally in the study of wave functions and oscillations, among other applications. When working with trigonometric identities, the secant function allows for certain simplifications, especially when dealing with products involving both sine and cosine.
Secant is particularly significant because it is frequently used in calculus. For example, expressing functions involving secant can simplify the integration of functions that might be difficult to handle if left in terms of sine and cosine. Understanding the utility of secant broadens your ability to tackle various mathematical problems.

Cosecant

The cosecant function is another reciprocal trigonometric function, similar to secant. It is defined as the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Cosecant is useful in contexts where sine values result in complex expressions, providing a simpler alternative.
In verifying identities, particularly those involving products like \( \sec \theta \csc \theta \), understanding that \( \csc \theta \) is simply \( \frac{1}{\sin \theta} \) allows for straightforward simplifications and manipulations in mathematical proofs.
This particular understanding helps not only in algebraic manipulations but also in recognizing symmetry and easing calculations involving trigonometric functions, especially when preparing for calculus-based problems.