91Ó°ÊÓ

Understanding trigonometric identities is crucial in solving calculus problems involving trigonometric functions. These identities are equations that relate different trigonometric functions. In part (a) of the exercise, we derive the identity \( \frac{\sec ^{2} x}{\tan x}=\frac{1}{\sin x \cos x} \). To achieve this, we first express secant \( \sec x \) and tangent \( \tan x \) in terms of sine \( \sin x \) and cosine \( \cos x \):

• \( \sec x = \frac{1}{\cos x} \)
• \( \tan x = \frac{\sin x}{\cos x} \)

Expanding \( \sec^2 x \) gives \( \left(\frac{1}{\cos x}\right)^2 = \frac{1}{\cos^2 x} \). Substituting and simplifying results in the identity \( \frac{1}{\cos x \sin x} \), verifying the trigonometric relationship. Recognizing these identities simplifies complex calculus problems, as they allow you to transform and manipulate functions into forms that are easier to integrate or differentiate.

Applying different integration techniques is an essential skill in calculus, especially when working with trigonometric integrals. In step 4 of the original solution, we use substitution to evaluate the integral \( \int \csc x \, dx \), transforming the expression into a form that is manageable. Start by rewriting the integral in terms of more commonly used functions and identities. Use:

• The identity \( \csc x = \frac{1}{\sin x} \)
• \( \sin 2x = 2 \sin x \cos x \)

Perform substitution with \( u = \cos x \) which transforms the derivative \( du = -\sin x \, dx \), simplifying the integral to \( \int \frac{-1}{u}\, du \). Practicing these techniques helps in dealing with complicated integrands that initially appear insurmountable. By recognizing patterns and applying the right method, such as substitution, integration by parts, or trigonometric identities, many problems can be simplified greatly.

Trigonometric integrals often involve integrals of functions like secant, cosecant, sine, or cosine. Each requires recognizing the structure and applying appropriate identities or substitution methods. In part (b), we evaluate \( \int \csc x \, dx \) using trigonometric identities:\
Set \( u = \cot x \) can simplify some trigonometric integrals into more recognizable patterns.

• The antiderivative \( \int \csc x \, dx \) is known as \( -\ln |\csc x + \cot x| + C \).

For part (c), evaluating \( \int \sec x \, dx \) involves using a different approach by setting \( u = \tan x \), so \( du = \sec^2 x \, dx \), which positions the integral for a straightforward solution. Recognizing the antiderivative as \( \ln |\sec x + \tan x| + C \), demonstrates the power of understanding these identities and transformations. Mastering these steps in solving trigonometric integrals, allows you to tackle various calculus problems with a structured approach.