Problem 98 - Precalculus Essentials

Question

Verify the identity: $$\frac{\sin (x-y)}{\cos x \cos y}+\frac{\sin (y-z)}{\cos y \cos z}+\frac{\sin (z-x)}{\cos z \cos x}=0$$

Step-by-Step Solution

Answer

The identity \(\frac{\sin (x-y)}{\cos x \cos y}+\frac{\sin (y-z)}{\cos y \cos z}+\frac{\sin (z-x)}{\cos z \cos x}=0\) is verified successfully by applying trigonometric identities and simplifying the expressions.

Step 1: Write down the identity

The identity to verify is: \(\frac{\sin (x-y)}{\cos x \cos y}+\frac{\sin (y-z)}{\cos y \cos z}+\frac{\sin (z-x)}{\cos z \cos x}=0\)

Step 2: Use the sine difference formula

The sine difference formula is \(\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)\). Apply this formula to each term in the identity equation: \(\frac{\sin x \cos y - \cos x\sin y}{\cos x \cos y}+\frac{\sin y \cos z - \cos y\sin z}{\cos y \cos z}+\frac{\sin z \cos x - \cos z \sin x }{\cos z \cos x}\)

Step 3: Simplify each term

Simplify each term by canceling the common factors in the numerators and denominators: \(\tan y - \tan x + \tan z - \tan y + \tan x - \tan z\)

Step 4: See the result

When you add up all the terms, they cancel each other out, giving a result of 0, which confirms the original identity

Key Concepts

Sine Difference Formula

The sine difference formula is a crucial identity in trigonometry that helps you deal with expressions involving the sine of two subtracted angles. This formula states: \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \). When applying this formula, you break down complex expressions involving angle differences into simpler sine and cosine components. This can make the manipulation and simplification of expressions more manageable. In the context of the exercise, this formula is used to transform each sine difference term:

• \(\sin(x-y)\) becomes \(\sin(x)\cos(y) - \cos(x)\sin(y)\)
• \(\sin(y-z)\) becomes \(\sin(y)\cos(z) - \cos(y)\sin(z)\)
• \(\sin(z-x)\) becomes \(\sin(z)\cos(x) - \cos(z)\sin(x)\)

After substituting these into the original identity, it becomes easier to see how terms can cancel out.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It's a vital tool for analyzing periodic phenomena such as waves, sound, and light. In trigonometry, we frequently encounter identities, like the sine and cosine difference formulas, which help simplify complex expressions and solve various mathematical problems. Key functions in trigonometry include:

• Sine \(\sin\)
• Cosine \(\cos\)
• Tangent \(\tan\)

These functions allow us to express relationships between angles and side lengths in a right triangle. Understanding these functions is essential for verifying identities, solving trigonometric equations, and understanding the properties of waves.

Angle Difference

The angle difference in trigonometry usually refers to the expression of one angle subtracted from another, such as \(x-y\), \(y-z\), or \(z-x\). Understanding angle differences is important for verifying trigonometric identities. When you have a difference between two angles, like \(x-y\), you can use identities such as the sine difference formula to break down and simplify the angle difference into sine and cosine components. These components help in evaluating complex trigonometric expressions by using common trigonometric identities:

• \(\sin(x-y)\)
• \(\cos(x-y)\)
• Expressing differences as: \(\tan(y) - \tan(x)\)

Using angle differences allows for efficient simplification and verification of trigonometric expressions and identities. By understanding how these differences are handled, you can better manage and solve problems involving multiple angles.