Problem 62 - Precalculus Essentials

Question

In Exercises \(59-68\), verify each identity. $$\cos ^{2} \frac{\theta}{2}=\frac{\sec \theta+1}{2 \sec \theta}$$

Step-by-Step Solution

Answer

The original identity \(\cos^2 (\frac{\theta}{2})=\frac{\sec \theta+1}{2 \sec \theta}\) has been verified by simplifying the right hand side using trigonometric identities and expressing it in the form on the left hand side.

Step 1: Identify the basic trigonometric identities

Recall some basic identities used frequently in trigonometry problems. These two will be very useful in this exercise: \(\cos^2x = 1/2(1+\cos2x)\) and \(\sec x = 1/\cos x\).

Step 2: Rewrite the right hand side of the equation using secant identity

We rewrite the right hand side of the equation, replacing \(\sec \theta\) with \(1/\cos \theta\). This gives: \(\frac{\sec \theta+1}{2 \sec \theta} = \frac{1/\cos\theta + 1}{2/(cos\theta)}\). Simplifying this gives \(\frac{1 + \cos\theta}{2}\).

Step 3: Express cos using another identity

We further express \(cos\theta\) using double angle identity for cosine: \( \cos\theta = 2\cos^2(\theta/2) - 1\). Applying this gives: \(\frac{1 + 2\cos^2(\theta/2) - 1}{2}\) which simplifies to \(\cos^2 (\theta/2)\).

Step 4: Verify the identity

So now we have established that the right hand side \(\frac{\sec \theta+1}{2 \sec \theta}\) simplifies to \(\cos^2 (\theta/2)\), which is the same as the left hand side. Thus, the original identity is verified.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec \theta \), is a crucial part of trigonometry. It is derived from the cosine function, with the relationship \( \sec \theta = \frac{1}{\cos \theta} \). This connection comes from the reciprocal nature of the cosine and secant. Whenever cosine is utilized, its secant counterpart might often become relevant.

Using the secant identity allows you to simplify expressions, especially in the context of verifying trigonometric identities. By substituting \( \sec \theta \) with \( \frac{1}{\cos \theta} \), we find alternative ways to approach and solve problems:

• For instance, converting given identities involving secant into expressions with cosine can make the simplification process more intuitive.
• This relationship is straightforward yet powerful, simplifying the manipulation of trigonometric identities.

Remember, secant function plays a key role not only in exercises but in various trigonometric applications.

Double Angle Identity

The double angle identities are vital when it comes to simplifying and solving trigonometric equations. For cosine, one of the double angle identities is \( \cos(2x) = 2\cos^2x - 1 \). This formula allows transformations between angles that differ by multiples, specifically doubling or halving angles.

These identities help when you are:

• Finding easier forms of expressions that appear difficult at first glance.
• Breaking down angles into simpler terms, making equations easier to solve.

In our current problem, the double angle identity helped express \( \cos \theta \) in terms of \( \cos^2(\theta/2) \). Hence, using this identity, we achieved the simplification needed to verify the trigonometric identity given in the exercise.

Trigonometric Verification

Trigonometric verification involves proving that a trigonometric equation or identity holds true for every allowable value of the variable. Verification isn't just about plugging numbers into equations; it's about transformation and simplification.

Key elements in this process:

• Identify and apply relevant trigonometric identities and formulas, as done with the secant and double angle identities.
• Rearrange terms to show equivalence of both sides of the equation. Choose strategic identities to effectively simplify your expressions.
• Simplify step-by-step, being mindful of arithmetic manipulations, to demonstrate the equality.

This exercise showcases a practical application—by simplifying both sides using known identities, we validate the correctness of the given identity. Verification strengthens your understanding of how trigonometric elements interact and ensures the soundness of mathematical results.