Problem 26 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\sin t}{\tan t}+\frac{\cos t}{\cot t}=\sin t+\cos t\)

Step-by-Step Solution

Answer

The exercise involves a trigonometric identity verification. On simplifying the left-hand side with suitable replacements, it was found that the LHS and the RHS are identical, thus confirming the given expression is a valid trigonometric identity.

Step 1: Use reciprocal identities to rewrite the LHS

Replace \(\tan t\) with \(\frac{\sin t}{\cos t}\) and \(\cot t\) with \(\frac{\cos t}{\sin t}\) on the left-hand side of the equation. This transforms the equation into \(\frac{\sin t}{\frac{\sin t}{\cos t}} + \frac{\cos t}{\frac{\cos t}{\sin t}}\).

Step 2: Simplify the fractions

Simplify each of the resulting complex fractions. This leads to simply \(\cos t + \sin t\), removing the complex fractions.

Step 3: Compare the simplifed LHS with the RHS

After simplification in step 2, the left-hand side of the identity \(\cos t + \sin t\) is same as the right-hand side. Thus, the original expression is a true identity.

Key Concepts

Simplifying Trigonometric Expressions

Trigonometry often involves expressions that, at first glance, might look intimidating. But fear not—simplification is like cleaning up a room, so every object finds its right place, leaving the scene neat and orderly. Simplifying trigonometric expressions primarily involves a series of substitutions and algebraic manipulations to convert a complex expression into a simpler, equivalent one.

Take the example of \(\frac{\sin t}{\tan t}+\frac{\cos t}{\cot t}\) from our exercise. The simplicity lies in understanding what trigonometric functions truly are and how they interact with each other. By breaking down \(\tan t\) into \(\frac{\sin t}{\cos t}\) and \(\cot t\) into \(\frac{\cos t}{\sin t}\), the initial expression morphs into a more recognisable and manageable form. Then, simplify complex fractions by multiplying the numerators and denominators, just as you would with any other fractions, until the complications melt away, revealing the simplified expression of \(\sin t + \cos t\).

Why Simplify Trigonometric Expressions?

Understanding or verifying the identities would be challenging without simplification. It's essential in solving trigonometry problems efficiently and is one of the fundamental skills in any mathematician's toolkit.

Reciprocal Trigonometric Functions

As the name suggests, reciprocal trigonometric functions are simply the 'flip' or reciprocal of the basic trigonometric functions. To visualize, think of flipping a pancake—rather than creating something new, you're just looking at the other side of what you already have. In trigonometry, this concept is essential.

For instance, the sine and cosine functions have their respective reciprocals, the cosecant (\(\csc\)) and secant (\(\sec\)), and the tangent function's reciprocal is the cotangent (\(\cot\)). It’s important to remember that \(\csc t = \frac{1}{\sin t}\), \(\sec t = \frac{1}{\cos t}\), and \(\cot t = \frac{1}{\tan t} = \frac{\cos t}{\sin t}\). These relationships are like the backbone of trigonometric identities; they allow you to express trig functions in different forms, which can be very useful in simplifying expressions and verifying identities.

Practical Example

In the context of our exercise, by recognising that \(\tan t\) and \(\cot t\) are the reciprocals of \(\cot t\) and \(\tan t\), respectively, the expression can be broken down further—leading to an elegant simplification.

Verifying Trigonometric Identities

Verifying trigonometric identities is akin to proving a point with solid evidence. It's a systematic way to demonstrate that two trigonometric expressions are equivalent under all conditions for which they are defined—no matter the value of the variable, the identity holds true. Verification is not just striking; it's convincing oneself or others that a trigonometric statement is true.

To verify an identity, you start by working with one side of the equation—usually the more complex side—transforming it step by step to match the other side. This process often involves simplifying expressions, employing reciprocal identities, and sometimes even leveraging other trigonometric identities, such as Pythagorean or co-function identities.

Breaking Down the Verification Process

In the step-by-step solution we have, the process started by using reciprocal identities on the left-hand side, continued with simplifying complex fractions, and ended with matching both sides. What seemed convoluted at first became clear, confirming that indeed, \(\frac{\sin t}{\tan t} + \frac{\cos t}{\cot t} = \sin t + \cos t\). Through verification, students sharpen their understanding and expand their ability to manipulate trigonometric expressions adeptly.