Problem 36 - Algebra and Trigonometry

Question

Prove the identity. $$ \cot (x+y)=\frac{\cot x \cot y-1}{\cot x+\cot y} $$

Step-by-Step Solution

Answer

The identity is proven by expressing both sides as \(\cot(x+y)\).

Step 1: Recall the Cotangent Addition Formula

The cotangent addition formula states that \[\cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b}.\]This is what we are asked to prove for the specific case when \(a = x\) and \(b = y\).

Step 2: Express Cotangent in Terms of Sine and Cosine

The cotangent function can be expressed in terms of sine and cosine as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). We need this expression to derive the identity.

Step 3: Start from the Right Side of the Identity

Let us start by expressing the right side of the given identity using the sine and cosine functions:\[\frac{\cot x \cot y - 1}{\cot x + \cot y} = \frac{\frac{\cos x}{\sin x}\cdot\frac{\cos y}{\sin y} - 1}{\frac{\cos x}{\sin x} + \frac{\cos y}{\sin y}}.\]

Step 4: Simplify Each Part Separately

We first find the numerator:\[\frac{\cos x \cos y}{\sin x \sin y} - 1 = \frac{\cos x \cos y - \sin x \sin y}{\sin x \sin y}.\]For the denominator:\[\frac{\cos x}{\sin x} + \frac{\cos y}{\sin y} = \frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y}.\]

Step 5: Combine the Expressions

The expression becomes:\[\frac{\frac{\cos x \cos y - \sin x \sin y}{\sin x \sin y}}{\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y}} = \frac{\cos x \cos y - \sin x \sin y}{\cos x \sin y + \cos y \sin x}.\]

Step 6: Use the Cosine and Sine Addition Formulas

Recall that the identity \(\cos(x+y) = \cos x \cos y - \sin x \sin y\) and \(\sin(x+y) = \sin x \cos y + \cos x \sin y\).This transforms the expression into:\[\frac{\cos(x+y)}{\sin(x+y)} = \cot(x+y).\]

Step 7: Conclusion

Since both sides simplify to \(\cot(x+y)\), the identity is verified.

Key Concepts

Cotangent Addition Formula

Understanding the Cotangent Addition Formula is crucial when dealing with trigonometric identities on addition or subtraction of angles. The formula states that:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
This formula allows us to find the cotangent of a sum of two angles using their individual cotangents. It’s especially handy because working directly with cotangent can be more straightforward in some cases compared to using sine or cosine.

• Why it Matters: Helps simplify expressions involving angle addition by reducing them into manageable parts. It is useful in solving equations and proving identities.
• Application: The formula is applied by substituting specific angle values into \( a \) and \( b \), which gives a pathway to proving identities like the one provided in the exercise.

Sine and Cosine Functions

The sine and cosine functions are fundamental building blocks of trigonometry. They are especially significant because all other trigonometric functions, such as tangent, cotangent, secant, and cosecant, are derived from them.
For an angle \( \theta \):

• Cosine: Represents the adjacent side divided by the hypotenuse in a right-angled triangle: \( \cos \theta = \frac{\text{Adjacent}}{{\text{Hypotenuse}}} \)
• Sine: Represents the opposite side divided by the hypotenuse: \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• Cotangent: Expressed in terms of sine and cosine as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

These relationships tell us how each trigonometric function is interconnected and provide a foundation for transformations and simplifications within many trigonometric identities, as evident in this exercise.

Trigonometric Simplification

Trigonometric simplification is the process of transforming a trigonometric expression into a more concise or manageable form. This skill is necessary for proving identities and solving equations more efficiently.
While simplifying, you often need to employ various trigonometric identities and functions equivalences, such as:

• Using Known Identities: For example, using \( \cos(x+y) = \cos x \cos y - \sin x \sin y \) and \( \sin(x+y) = \sin x \cos y + \cos x \sin y \) can greatly simplify expressions.
• Combining and Reducing: Combine fractions and reduce complex expressions using common denominators and algebraic manipulations.
• Converting Functions: Use the relationships between sine, cosine, and other trigonometric functions to interchange and simplify terms.

Mastering these techniques is key to tackling trigonometric proofs and is showcased as the identity \( \cot(x+y) \) is proved in the original exercise.