Problem 17 - Algebra and Trigonometry

Question

Verify each identity. \(\sin t \tan t=\frac{1-\cos ^{2} t}{\cos t}\)

Step-by-Step Solution

Answer

The given trigonometric identity is correct. Both sides of the equation simplify to \(\frac{\sin^2 t}{\cos t}\), verifying the identity.

Step 1: Rewrite the Trigonometric function in the terms of sine and cosine

We can express \(\sin t \tan t\) and \(\frac{1-\cos ^{2} t}{\cos t}\) as \(\sin t \frac{\sin t}{\cos t}\) and \(\frac{1-\cos ^{2} t}{\cos t}\), respectively. Thus, the equation can now be written as \(\sin t \frac{\sin t}{\cos t} = \frac{1-\cos ^{2} t}{\cos t}\)

Step 2: Simplify and Factor out

On simplifying the left side of the equation, we have \(\frac{\sin^2 t}{\cos t}\). Let us simplify the right side as well. Since \(1-\cos^2 t\) is equivalent to \(\sin^2 t\) from the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\), we can rewrite the right side as \(\frac{\sin^2 t}{\cos t}\)

Step 3: Match both sides

Now both sides of the equation are identical. They are both \(\frac{\sin^2 t}{\cos t}\), therefore proving that \(\sin t \tan t = \frac{1-\cos ^{2} t}{\cos t}\)

Key Concepts

Sine and Cosine Functions

When diving into the subject of trigonometry, two significant functions that we come across are the sine and cosine functions. Sine, typically denoted as \( \sin \theta \), and cosine, denoted as \( \cos \theta \), are fundamental trigonometric functions. They relate the angles of a right triangle to the lengths of its sides.
Here are a few key points to understand about these functions:

• The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

The unit circle can also be used to define these functions in a broader sense, allowing angles greater than 90 degrees. The coordinates of any point on the unit circle correspond to \( (\cos \theta, \sin \theta) \).
In trigonometric identities, expressing more complex functions in terms of sine and cosine can simplify the problem-solving process. For example, the tangent function \( \tan \theta \) can be rewritten as \( \frac{\sin \theta}{\cos \theta} \). By converting other trig functions into sine and cosine, it often becomes easier to manipulate and solve trigonometric equations.

Pythagorean Identity

The Pythagorean identity is a critical component in the study of trigonometry. It is an identity that provides a relationship between sine and cosine functions of any angle. The main form of the Pythagorean identity is:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity comes from the Pythagorean theorem in geometry and is incredibly useful in simplifying trigonometric expressions. For instance, when you have an expression like \(1 - \cos^2 \theta\), you can substitute \( \sin^2 \theta \) from the identity.
This conversion is crucial in the verification of trigonometric identities, as seen in the original exercise. Here, substituting \( \sin^2 t \) for \( 1 - \cos^2 t \) makes it straightforward to match two sides of an equation, allowing for verification of complex identities.
Remember:

• Use the Pythagorean identity to simplify expressions involving \( \sin^2 \theta \) and \( \cos^2 \theta \).
• This identity is valid for any angle \( \theta \), allowing for broad applications.

Trigonometric Simplification

Trigonometric simplification refers to the process of making a complex trigonometric expression more manageable or reducing it to a simpler form. This process uses various trigonometric identities and properties to achieve equivalency between expressions.Here's what to focus on:

• Rewrite complex terms using basic trigonometric identities, like expressing \( \tan \theta \) as \( \frac{\sin \theta}{\cos \theta} \).
• Use the Pythagorean identity to eliminate terms or transform them into a simpler form.
• Aim to reduce expressions to the most basic terms possible, like \( \sin^2 \theta = 1 - \cos^2 \theta \).

Simplification helps in solving, verifying, and understanding complex trigonometric expressions. For instance, by converting both sides of the equation into similar forms, comparing and equating them becomes easier, facilitating a direct proof of identity.
Remember that practice is key. Regularly working through such simplifications deepens understanding and enhances your ability to manipulate trigonometric equations efficiently.