Problem 53 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{1+\cos \phi}{\sin \phi}=\frac{\sin \phi}{1-\cos \phi} $$

Step-by-Step Solution

Answer

The identity is verified, as both sides simplify to \(\sin^2 \phi\).

Step 1: Cross Multiply

First, cross-multiply both sides to eliminate the fractions. This gives us:\[(1 + \cos \phi)(1 - \cos \phi) = (\sin \phi)(\sin \phi)\]

Step 2: Expand and Simplify the Left Side

Now, expand the expression \((1 + \cos \phi)(1 - \cos \phi)\), which is a difference of squares:\[1 - \cos^2 \phi\]

Step 3: Use Pythagorean Identity

Recall the Pythagorean identity that \(1 - \cos^2 \phi = \sin^2 \phi\). Substitute this into the equation:\[\sin^2 \phi = \sin^2 \phi\]

Step 4: Verify the Equality

Since both sides simplify to \(\sin^2 \phi\), the original identity holds true.

Key Concepts

Cross Multiplication

Cross multiplication is a useful tool for solving equations that involve fractions. It works by multiplying each numerator by the opposite denominator, effectively getting rid of the fractions. This makes the equation easier to handle and solve. In our exercise, we applied cross multiplication to the fractions in the trigonometric identity:

• The left side: \( (1 + \cos \phi) \)
• The right side: \( \sin \phi \)

We cross-multiplied these terms, which resulted in a new equality without fractions:\[(1 + \cos \phi)(1 - \cos \phi) = (\sin \phi)(\sin \phi)\]This method is particularly useful in trigonometry when simplifying and verifying identities, as it eliminates complex-looking fraction problems for easier manipulation.

Difference of Squares

The difference of squares is a specific algebraic technique for factoring expressions in the form \( a^2 - b^2 \). This expression can be factored into:\[(a + b)(a - b)\]In our equation,

• \(a = 1\)
• \(b = \cos \phi\)

We use this technique to expand \((1 + \cos \phi)(1 - \cos \phi)\), which simplifies into a difference of squares:\[1 - \cos^2 \phi\]The neat aspect of this is that it reduces complicated expressions into simpler forms that are more manageable, which aids greatly in solving and proving identities like the one we examined in our exercise.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that states:\[\sin^2 \phi + \cos^2 \phi = 1\]It's called this because it echoes the Pythagorean theorem, which links the sides of a right triangle. From this identity, we can derive:\[1 - \cos^2 \phi = \sin^2 \phi\]We used this derived expression to simplify our equation after using the difference of squares technique. By substituting\[1 - \cos^2 \phi = \sin^2 \phi\]into our equation, we established the given trigonometric identity:\[ \sin^2 \phi = \sin^2 \phi \]Recognizing and applying the Pythagorean identity is crucial when working with trigonometric equations and verifications, as it frequently provides the simplest pathway to solutions.