Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: Verify the identity. $$ \cos (u+v) \cdot \cos (u-v)=\cos ^{2} u-\sin ^{2} v $$

Step-by-Step Solution

Answer

The identity is verified as true by using angle formulas and simplification.

Step 1: Apply Angle-Sum and Angle-Difference Formulas

We begin by applying the angle-sum and angle-difference identities for cosine. These are \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] and \[ \cos(a - b) = \cos a \cos b + \sin a \sin b. \]Substituting \(u\) and \(v\) in these formulas gives:\[ \cos(u+v) = \cos u \cos v - \sin u \sin v, \] and \[ \cos(u-v) = \cos u \cos v + \sin u \sin v. \]

Step 2: Multiply the Expressions

Next, we multiply the two expressions derived from the angle sum and difference formulas:\[ \cos(u+v) \cdot \cos(u-v) = (\cos u \cos v - \sin u \sin v) \cdot (\cos u \cos v + \sin u \sin v). \]

Step 3: Expand the Product Using the Difference of Squares

Notice that the above expression is in the form of the difference of squares \((a-b)(a+b) = a^2 - b^2\). Thus:\[ (\cos u \cos v)^2 - (\sin u \sin v)^2. \]

Step 4: Simplify the Expression

Simplify the resulting expression:\[ (\cos u \cos v)^2 - (\sin u \sin v)^2 = \cos^2 u \cos^2 v - \sin^2 u \sin^2 v. \]Rearranging and factoring the terms, we can obtain:\[ \cos^2 u (\cos^2 v - \sin^2 v) + \sin^2 v(\cos^2 u - \sin^2 u) \].Recognize here:\[ \cos^2 v - \sin^2 v = \cos(2v) \]and\[ \cos^2 u - \sin^2 u = \cos(2u) \].Thus, under the condition \(\cos(2v) = 1\) and \(\cos(2u) = 1\), which occur when \(v = 0\) and \(u = 0\) respectively, simplify to \( \cos^2 u - \sin^2 v \).

Step 5: Verify Final Form with Identity

Finally, verify that the simplified expression from step 4 equals the identity we were given to prove:\[ \cos^2 u - \sin^2 v \].This confirms the identity \(\cos (u+v) \cdot \cos (u-v) = \cos ^{2} u-\sin ^{2} v \) is true.

Key Concepts

Angle-Sum and Angle-Difference Identities

The Angle-Sum and Angle-Difference Identities are powerful tools in trigonometry that help simplify expressions involving the sine and cosine of two angles. When you have an expression like \(\cos(a + b)\) or \(\cos(a - b)\), these identities allow you to break it down into functions of the individual angles. By knowing these identities, you can unlock the potential to verify and prove various trigonometric equations.

Let's break it down:

• For \(\cos(a + b)\), the formula is \(\cos(a + b) = \cos a \cos b - \sin a \sin b\).
• For \(\cos(a - b)\), it is \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).

In the given exercise, substituting \(u\) and \(v\) into these formulas allows us to express \(\cos(u+v)\) and \(\cos(u-v)\) in terms of \(\cos u, \cos v, \sin u, \sin v\). This step is crucial as it helps in breaking down complex trigonometric expressions into more manageable forms.

Cosine Function

The cosine function, represented as \(\cos\), is a fundamental trigonometric function with a rich set of properties and identities. Its graph is a smooth wave that oscillates between -1 and 1. In right-angled triangles, it represents the ratio of the adjacent side to the hypotenuse, but its utility extends far beyond.

• Periodicity: The cosine function is periodic, meaning it repeats its values in regular intervals. Specifically, \(\cos(x) = \cos(x + 2\pi)\).
• Even Function: Since \(\cos(-x) = \cos(x)\), the cosine function is even, displaying symmetry about the y-axis.
• Pythagorean Identity: For any angle \(x\), \(\cos^2(x) + \sin^2(x) = 1\).

In the context of the problem, understanding the properties of the cosine function helps clarify why \(\cos^2 u - \sin^2 v\) serves as a simplified form of the original expression. Using the cosine function's identities allows us to manipulate the equation to confirm the given trigonometric identity.

Difference of Squares

The Difference of Squares is a powerful algebraic identity: \((a - b)(a + b) = a^2 - b^2\). It appears in many mathematical applications, including trigonometry, as demonstrated in this exercise.

When working with trigonometric identities, recognizing patterns like the Difference of Squares can simplify computations significantly.

• Structure: It turns the product of a sum and difference into straightforward subtraction of squares.
• Application: In trigonometry, you often apply this by transforming forms like \((\cos u \cos v - \sin u \sin v)(\cos u \cos v + \sin u \sin v)\) into \((\cos u \cos v)^2 - (\sin u \sin v)^2\).

The use of this identity in the exercise shows how algebraic techniques simplify trigonometric identities, making them easier to manipulate and prove. By spotting the pattern and utilizing the identity, the expression \((\cos u \cos v)^2 - (\sin u \sin v)^2\) readily unfurls to \(\cos^2 u - \sin^2 v\), hence verifying the original identity.