Problem 39 - Calculus Single Variable

Question

Use one or more of the basic trigonometric identities to derive the given identity. \(\cos (\theta) \cos (\phi)=\frac{\cos (\theta-\phi)+\cos (\theta+\phi)}{2}\)

Step-by-Step Solution

Answer

The trigonometric identity is verified as correct.

Step 1: Recall the Cosine Addition and Subtraction Formulas

First, we need to remember the cosine addition and subtraction formulas for angles. The formulas are as follows:- For cosine addition: \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\).- For cosine subtraction: \(\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)\).

Step 2: Write the Given Identity

We are given the identity \(\cos(\theta) \cos(\phi) = \frac{\cos(\theta-\phi) + \cos(\theta+\phi)}{2}\) and need to verify it using basic trigonometric identities.

Step 3: Express the Right-Hand Side Using Cosine Formulas

Substitute the formulas for cosine of addition and subtraction into the right-hand side of the given identity:\[\frac{\cos(\theta-\phi) + \cos(\theta+\phi)}{2} = \frac{\left(\cos(\theta)\cos(\phi) + \sin(\theta)\sin(\phi)\right) + \left(\cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi)\right)}{2}\]

Step 4: Simplify the Expression

Combine like terms from the expansion:\[= \frac{\cos(\theta)\cos(\phi) + \sin(\theta)\sin(\phi) + \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi)}{2}\]The \(\sin(\theta)\sin(\phi)\) terms cancel each other out, leaving:\[= \frac{2\cos(\theta)\cos(\phi)}{2}\]

Step 5: Final Simplification

Divide both the numerator and the denominator by 2:\[= \cos(\theta)\cos(\phi)\]

Step 6: Conclusion: Verify the Identity

We have derived that the right-hand side simplifies to the left-hand side, \(\cos(\theta)\cos(\phi)\), confirming the identity is true.

Key Concepts

Cosine Addition Formula

The cosine addition formula is an essential trigonometric identity that allows us to find the cosine of the sum of two angles. This formula is expressed as:\[\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\]Here’s what it means:

• We take two angles, say \(a\) and \(b\), and want to know the cosine of their sum, \(a + b\).
• Instead of finding this sum directly, we can express it as a combination of simpler trigonometric functions: the product of cosines and the product of sines of these angles.
• This breaks down the problem into easier parts since cosine and sine of individual angles might be known.

The usage of this formula becomes particularly helpful in proving identities and solving trigonometric equations. By applying this formula, we can transform a complex addition problem into a simpler one involving multiplications and subtractions. This is evident in the identity discussed where adding the angles \(\theta\) and \(\phi\) is simplified using the formula.

Cosine Subtraction Formula

Just like the addition formula, we have the cosine subtraction formula. This formula gives us the cosine of the difference between two angles:\[\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)\]Here's how it simplifies our trigonometric calculations:

• This formula is essentially the opposite of the addition formula; the main difference lies in the sign of the sine terms.
• When subtracting two angles \(a\) and \(b\), this identity lets us break down the angle difference into products of cosines and sines.
• It's particularly handy when dealing with angle differences, making it easier to calculate unfamiliar angles using known ones.

In the given trigonometric identity, the subtraction formula allows us to represent \(\cos(\theta - \phi)\) in a form that is symmetrical to the addition scenario. This symmetry and parallelism are crucial when verifying the identity as shown in the original step-by-step solution.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions to simpler forms. This is often done using trigonometric identities and algebraic manipulation. Here’s how it applies to our exercise:

• We start with a complex expression or identity that involves multiple trigonometric functions.
• By substituting known identities, like the cosine addition and subtraction formulas, we turn this expression into an application of these basic identities.
• The next step is to perform algebraic operations, such as combining like terms or canceling terms across the numerator and denominator.

In our example exercise, the initially complicated expression \(\frac{\cos(\theta-\phi) + \cos(\theta+\phi)}{2}\) was simplified by substituting in the formulas for cosine of angles, resulting in the cancellation of \(\sin(\theta)\sin(\phi)\) terms. The expression reduced to \(\cos(\theta)\cos(\phi)\), confirming the given identity. Simplifying these expressions not only proves identities but also offers a robust framework for tackling more advanced trigonometric problems.