Problem 46 - Contemporary Precalculus

Question

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$\frac{\sec ^{2} t \csc t}{\csc ^{2} t \sec t}$$

Step-by-Step Solution

Answer

Question: Simplify the given expression \(\frac{\sec ^{2} t \csc t}{\csc ^{2} t \sec t}\). Answer: \(1\)

Step 1: Write the Function in Terms of Sine and Cosine

To simplify the expression, we can write the \(\sec\) and \(\csc\) in terms of basic trigonometric functions, i.e., sine and cosine. Remember that \(\sec t = \frac{1}{\cos t}\) and \(\csc t = \frac{1}{\sin t}\). So our expression becomes: $$\frac{\left(\frac{1}{\cos^2 t}\right) \left(\frac{1}{\sin t}\right)}{\left(\frac{1}{\sin^2 t}\right) \left(\frac{1}{\cos t}\right)}$$

Step 2: Simplify the Expression

Now, let's simplify the expression by cancelling common factors in the numerator and denominator. $$\frac{\frac{1}{\sin t \cos^2 t}}{\frac{1}{\sin^2 t \cos t}}$$ To simplify further, multiply both the numerator and the denominator by \((\sin t \cos^2 t)(\sin^2 t \cos t)\). $$\frac{(1) \cdot (\sin t \cos^2 t)(\sin^2 t \cos t)}{(\sin t \cos^2 t) \cdot (\sin^2 t \cos t)}$$ Now, cancel out the common factors: $$\frac{\sin t \cos^2 t \sin^2 t \cos t}{\sin t \cos^2 t \sin^2 t \cos t} = 1$$ So, our simplified expression is: $$\boxed{1}$$

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved, where that equation is defined. They are the backbone of trigonometry and essential tools for simplifying expressions. Some of the most important identities include the Pythagorean identities, reciprocal identities, and angle sum/difference identities.

• Reciprocal Identities: These show how one trigonometric function can be expressed as the reciprocal of another. For instance, \(\sec t = \frac{1}{\cos t}\) and \(\csc t = \frac{1}{\sin t}\).
• Pythagorean Identities: Perhaps the most famous, \(\sin^2 \theta + \cos^2 \theta = 1\). These are useful for converting between sine and cosine.

Understanding these identities allows you to transform and simplify trigonometric expressions. This skill is useful in higher-level mathematics, physics, and engineering.

Sine and Cosine

When working with trigonometric simplifications, sine and cosine are often the functions of choice. They are the foundational waveforms of trigonometry and define relationships between angles in right triangles.

• Sine Function: The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. It's a periodic function that helps in wave analysis.
• Cosine Function: Similarly, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. Like sine, it measures the extent of the wave.

These functions are also used in reciprocal identities, making them crucial for expressing more complex trigonometric functions in simpler forms. Many trigonometric problems become easier by using sine and cosine.

Trigonometric Simplification

Simplifying trigonometric expressions may seem complex, but it boils down to substituting complicated parts with more straightforward equivalents using identities.

The process usually involves the following steps:

• Expressing in terms of Sine and Cosine: This is often the first step. By expressing all parts of the expression in terms of sine and cosine, it becomes easier to spot common factors and cancel them.
• Cancelling and Reducing: After substituting, the next step is to cancel out derivatives and reduce the expression to its simplest form. Common factors in the numerator and denominator are typically simplified using elementary arithmetic.

For example, in our exercise, converting \(\sec\) and \(\csc\) into \(1/\cos\) and \(1/\sin\) respectively allowed the expression to be simplified to just \(1\) by cancelling like terms. Simplification not only makes expressions manageable but also helps solve trigonometric equations efficiently.