Problem 91 - Precalculus

Question

For the following exercises, prove the identities provided. $$ \frac{\cos (a+b)}{\cos a \cos b}=1-\tan a \tan b $$

Step-by-Step Solution

Answer

Proven: \( \frac{\cos (a+b)}{\cos a \cos b} = 1 - \tan a \tan b \).

Step 1: Start with the Left Side (LS)

We begin with the left side of the equation: \[\text{LS} = \frac{\cos(a+b)}{\cos a \cos b}\]

Step 2: Expand the Cosine of a Sum

Use the cosine angle sum identity: \[\cos(a+b) = \cos a \cos b - \sin a \sin b\]Substitute this into the LS:\[\frac{\cos(a+b)}{\cos a \cos b} = \frac{\cos a \cos b - \sin a \sin b}{\cos a \cos b}\]

Step 3: Split the Fraction

Separate the fraction into two parts:\[\frac{\cos a \cos b}{\cos a \cos b} - \frac{\sin a \sin b}{\cos a \cos b}\]This simplifies to:\[1 - \frac{\sin a \sin b}{\cos a \cos b}\]

Step 4: Simplify the Second Term

The term \(\frac{\sin a \sin b}{\cos a \cos b}\) can be rewritten using the definition of tangent:\[\frac{\sin a}{\cos a} \cdot \frac{\sin b}{\cos b} = \tan a \tan b\]

Step 5: Combine the Results

Substituting back into the expression, we have:\[1 - \tan a \tan b\]This matches the Right Side (RS) of the equation.

Key Concepts

Cosine of a Sum

To understand the solution to the given exercise, it helps to first revisit the cosine of a sum identity. This identity is a fundamental tool in trigonometry, used to simplify the cosine of the sum of two angles, say \(a\) and \(b\). Often taught early in trigonometry courses, it is expressed as:

• \(\cos(a+b) = \cos a \cos b - \sin a \sin b\)

This formula shows how the cosine of an angle sum is actually the difference of two products: one involving cosines and the other involving sines. This identity is essential for proving many trigonometric identities because it allows us to break down more complicated expressions.
In our particular problem, the identity is used to express \(\cos(a+b)\) in a way that makes it possible to simplify the original fraction and reveal the relationship between different trigonometric functions. This step is the key to moving forward with further manipulations in trigonometric proofs.

Tangent

The tangent function, often abbreivated as \(\tan\), is another fundamental trigonometric function important for this problem. Tangent is defined as the ratio of the sine to the cosine of an angle:

• \(\tan a = \frac{\sin a}{\cos a}\)

This definition helps us rewrite expressions involving sines and cosines in terms of tangent, making it easier to manage trigonometric identities.
In the given exercise, one step requires replacing \(\frac{\sin a}{\cos a} \cdot \frac{\sin b}{\cos b}\) with \(\tan a \tan b\). By using this definition of tangent, the expression becomes much simpler and easier to match with the terms of the identity we want to prove.
Understanding how transformations between sine, cosine, and tangent work is crucial when tackling trigonometric proofs, as you'll frequently need to switch between these forms.

Trigonometric Proofs

Trigonometric proofs often involve showing that two different-looking expressions are actually equivalent, using trigonometric identities and algebraic manipulation. They require:

• Familiarity with basic identities such as the sine, cosine, and tangent definitions.
• Skills in algebra to simplify and rearrange expressions.
• Attention to detail to ensure each step follows logically from the previous one.

In our exercise, the goal was to prove that \(\frac{\cos(a+b)}{\cos a \cos b} = 1 - \tan a \tan b\). Starting from the left side given expression, we used the cosine of a sum identity to express \(\cos(a+b)\) in terms of \(\cos a\), \(\cos b\), \(\sin a\), and \(\sin b\).
Next, we separated the fractions and simplified using the definition of tangent. This step-by-step manipulation helps match the initial seeming different expressions eventually leading to the right side of the equation. Mastering such proofs helps strengthen your understanding of the relationships between trigonometric functions.