Question

Verify the identity. \(\sec t-\cos t=\tan t \sin t\)

Short answer

To verify the trigonometric identity \(\sec t-\cos t=\tan t \sin t\), we rewrite the secant and tangent functions in terms of sine and cosine and simplify the expression. After a series of manipulations, we are able to establish that both sides of the equation are equal, confirming the identity: \[ \frac{\sin^2 t}{\cos t} = \frac{(\sin t)(\sin t)}{\cos t} \] Hence, we have verified the identity \(\sec t - \cos t = \tan t \sin t\).

Step-by-step solution

Rewrite the given identity using sine and cosine functions

We will rewrite the secant and tangent functions in terms of sine and cosine to simplify the manipulation of the equation. We know that \(\sec t = \frac{1}{\cos t}\) and \(\tan t = \frac{\sin t}{\cos t}\). So we rewrite the equation as follows: \[ \frac{1}{\cos t} - \cos t = \frac{\sin t}{\cos t} \cdot \sin t \]

Simplify the left side of the equation

To simplify the left side of the equation, we will find a common denominator, which is \(\cos t\). Then we will combine the fractions: \[ \frac{1}{\cos t} - \frac{\cos^2 t}{\cos t} = \frac{(\sin t)(\sin t)}{\cos t} \]

Combine fractions on the left side

Now that we have a common denominator of \(\cos t\), we can combine the fractions on the left side of the equation: \[ \frac{1 - \cos^2 t}{\cos t} = \frac{(\sin t)(\sin t)}{\cos t} \]

Use the Pythagorean identity to simplify the left side

We will use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) to rewrite the left side of the equation. Since we have \(1 - \cos^2 t\), we will rearrange the Pythagorean identity to get \(\sin^2 t = 1 - \cos^2 t\). This simplifies the expression as follows: \[ \frac{\sin^2 t}{\cos t} = \frac{(\sin t)(\sin t)}{\cos t} \]

Compare both sides of the equation

Finally, let's take a closer look at both sides of the equation: \[ \frac{\sin^2 t}{\cos t} = \frac{(\sin t)(\sin t)}{\cos t} \] We can see that both sides of the equation are the same, which means we have successfully verified the identity: \[ \sec t - \cos t = \tan t \sin t \]

Key concepts

Secant Function

The secant function, denoted as \textbf{sec}, is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, which means it is defined as \( \sec x = \frac{1}{\cos x} \) where \( x \) is the angle. Though less common than the sine and cosine, the secant function plays a significant role in trigonometry, often appearing in various mathematical and real-world contexts such as calculus and physics.

When dealing with trigonometric identities, understanding the relationship between secant and cosine is crucial. For instance, when trying to verify an identity like \( \sec t - \cos t = \tan t \sin t \), it helps to express all terms in sine and cosine so that other identities, such as the Pythagorean identity, can be applied to simplify and verify the equation.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that is derived from the Pythagorean Theorem applied to the unit circle. It states that for any angle \( t \), \( \sin^2 t + \cos^2 t = 1 \). This identity illustrates the inherent connection between sine and cosine functions and establishes the groundwork for solving more complex trigonometric equations.

To utilize the Pythagorean identity, one might need to rearrange it depending on which trigonometric function one is solving for. For example, to find \( \sin^2 t \) in terms of \( \cos t \), you can rewrite the identity as \( \sin^2 t = 1 - \cos^2 t \). This technique is illustrated in the exercise, where this form of the Pythagorean identity simplifies the expression significantly and leads to verifying the given trigonometric identity.

Sine and Cosine

The sine and cosine functions are foundational in trigonometry, representing the ratios of sides within a right triangle relative to a specific angle or on the unit circle. Sine, denoted as \(\textbf{sin}\), is defined as the ratio of the opposite side to the hypotenuse of a right triangle, while cosine, denoted as \(\textbf{cos}\), is the ratio of the adjacent side to the hypotenuse.

Beyond triangles, these functions are also used to model periodic phenomena like sound waves and tides. They are essential in the verification process of trigonometric identities, as seen in the exercise, where manipulating these functions and applying their relations, such as the Pythagorean identity, allows for simplification and verification of trigonometric equations. Understanding how to manipulate these functions is vital in verifying identities like \( \sec t - \cos t = \tan t \sin t \).