Problem 47 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\sec t+1}{\tan t}=\frac{\tan t}{\sec t-1}\)

Step-by-Step Solution

Answer

Successfully verified the trigonometric identity: \(\frac{\sec t+1}{\tan t} = \frac{\tan t}{\sec t-1}\). By converting the original expressions into sine and cosine, and then simplifying the resulting expressions, it was demonstrated that the two sides of the equation are in fact equal.

Step 1: Express in terms of sine and cosine

Let's first convert all the trigonometric expressions into sine and cosine since they are the primary trigonometric functions. The secant of t, \(sec t\) is the reciprocal of the cosine of t, or \(sec t = \frac{1}{cos t}\) and the tangent of t, \(tan t\) can be expressed as the ratio of sine of t to cosine of t, or \(tan t = \frac{sin t}{cos t}\). So, substitute these values into the given equation we have:\(\frac{1/cos t + 1}{sin t / cos t} = \frac{sin t / cos t}{1/cos t - 1}\)

Step 2: Simplify both sides

We can simplify this expression by multiplying everything by \(cos t\), to eliminate the fractions:\(\frac{1 + cos t}{sin t} = \frac{sin t}{1 - cos t}\)Then, multiply both sides of the equation by \(sin t*(1-cos t)\):\((1 + cos t) * (1 - cos t) = sin^2 t\)

Step 3: Apply Pythagorean identity

We know that one of the primary Pythagorean identities in trigonometry is \(sin^2 t + cos^2 t = 1\).\nSo, replace \(sin^2 t\) with \(1 - cos^2 t\) in the equation:\((1 - cos^2 t) = (1 - cos^2 t)\)

Step 4: Verification

The equation obtains an identity after the substitutions, meaning that the left side of the equation is equal to the right side, hence verifying the given trigonometric identity.

Key Concepts

Sine and Cosine Transformation

Trigonometric identities often involve transforming expressions using sine and cosine, the fundamental trigonometric functions. This transformation is essential as it simplifies complex identities and makes understanding deeper relationships in trigonometry more accessible. Let's go through it step by step:

• Secant, known as \(\sec t\), is the reciprocal of cosine and is expressed as \(\frac{1}{\cos t}\). This means that when we see secant in an identity, we can replace it with its cosine equivalent.
• Tangent, represented by \(\tan t\), can be defined as the ratio of sine to cosine, \(\frac{\sin t}{\cos t}\). This transformation helps to express tan in terms of both sine and cosine.

By switching secant and tangent into these sine and cosine forms, we are just re-expressing the same relationship in a way that is easier to manipulate. These transformations are like switching languages to understand a foreign text better.
However, the meanings remain the same, just conveyed differently.

Pythagorean Identities

The Pythagorean identities are some of the most fundamental identities in trigonometry. They are critical tools for simplifying and verifying trigonometric expressions. Here's why they're so powerful:

• The most well-known identity is \(\sin^2 t + \cos^2 t = 1\). It's like the trigonometric version of the Pythagorean theorem, which shows the intrinsic relationship between sine and cosine.
• This identity lets you replace expressions like \(\sin^2 t\) with \(1 - \cos^2 t\), and vice versa, where applicable. Such replacements simplify the trigonometric forms.

By using these identities, you can reduce complex trigonometric expressions to their simplest forms. Plus, it serves as a check to ensure two sides of an equation can be shown to be identical, verifying complex trigonometric identities.

Trigonometry Verification Steps

Verifying trigonometric identities might seem complex, but when broken down into systematic steps, it becomes much simpler. Let's look at how this is done effectively:

• Needs Simplification: Initially, express all complex trigonometric terms using basic identities, like sine and cosine. This not only simplifies the terms but makes comparisons straightforward.
• Eliminate Fractions: Multiply through by the denominators where necessary to eliminate fractions. This simplifies comparisons between the two sides of the equation.
• Apply Identities: Use identities, such as the Pythagorean identity, to simplify each side. This often enables you to find common grounds between the two sides of a proposed identity.

Think of these steps as a checklist that ensures you've methodically processed each part of the equation. This provides logical clarity and confidence in confirming the trigonometric identities.