Problem 25 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\sin t}{\csc t}+\frac{\cos t}{\sec t}=1\)

Step-by-Step Solution

Answer

After replacing the reciprocal identities and simplifying the equation, we can apply Pythagorean identity which proves the original equation is correct.

Step 1: Replace the reciprocal identities

The reciprocal identity of \(\csc t\) is \(\frac{1}{\sin t}\) and the reciprocal identity of \(\sec t\) is \(\frac{1}{\cos t}\). Replace these in the original equation: \(\frac{\sin t}{\frac{1}{\sin t}} + \frac{\cos t}{\frac{1}{\cos t}}\)

Step 2: Simplify the equation

By simplifying the equation, we get: \( \sin t*\sin t + \cos t * \cos t\) which results into: \( \sin^{2} t+ \cos^{2} t\)

Step 3: Apply trigonometric identity

Recall that one of the fundamental trigonometric identities is the Pythagorean identity: \( \sin^{2} t+ \cos^{2} t = 1\). Therefore, the left-hand side of our equation simplifies to 1

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry, much like the Pythagorean Theorem is in geometry. It states that for any angle \(t\), the squares of the sine and cosine functions will add up to 1: \[ \sin^2 t + \cos^2 t = 1 \] This identity is immensely useful, as it relates the fundamental trigonometric functions to one another and allows us to simplify expressions. In geometry, this identity can be visualized using the unit circle, where the radius is 1 and any point on the circle satisfies this equation. Understanding and memorizing this identity helps simplify many trigonometric problems, ensuring you can transform complex expressions into more manageable forms. Once you've mastered using it, you'll find it enables quick and efficient problem-solving. For instance, as shown in our exercise, applying the Pythagorean Identity transitioned the left side of the equation into a simple and elegant "1".

Reciprocal Identities

Reciprocal Identities are another key group of relationships in trigonometry that make it easier to manipulate and solve equations. These identities involve relationships between common trigonometric functions and their respective reciprocals:

• The reciprocal of sine is cosecant: \( \csc t = \frac{1}{\sin t} \)
• The reciprocal of cosine is secant: \( \sec t = \frac{1}{\cos t} \)
• The reciprocal of tangent is cotangent: \( \cot t = \frac{1}{\tan t} \)

By recognizing these identities, simplifying equations becomes significantly easier. You can substitute reciprocal identities to transform potentially unwieldy functions into simpler expressions. For the exercise, the reciprocals \( \csc t \) and \( \sec t \) were converted, making the simplification straightforward. This step allowed us to connect directly to the more simplified Pythagorean Identity, showcasing the power of these reciprocal transformations.

Trigonometric Simplification

Trigonometric Simplification involves using identities to reduce complex expressions to simpler ones. This not only makes calculations easier but also deepens understanding of relationships between trigonometric functions. In practice, this often means recognizing when to apply specific identities, like the Pythagorean or Reciprocal Identities, to transform one expression into another. In our example problem, simplifications were performed by replacing \( \csc t \) and \( \sec t \) using reciprocal identities, and subsequently by employing the Pythagorean identity to simplify down to the integer 1. Here are some tips for trigonometric simplification:

• Always check for known identities that relate existing parts of your equation.
• Look for opportunities to factor or combine like terms.
• Keep your final goal in mind, simplifying in steps that edge closer to a recognizably simpler form or known identity.

The more you practice, the more patterns you will notice, making simplification feel intuitive.