Problem 90 - Algebra and Trigonometry

Question

Verify the identity. $$ (\sin \alpha-\tan \alpha)(\cos \alpha-\cot \alpha)=(\cos \alpha-1)(\sin \alpha-1) $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \((\sin \alpha - 1)(\cos \alpha - 1)\).

Step 1: Expand the Left Hand Side

To verify the given identity, begin by expanding the left-hand side (LHS) expression \((\sin \alpha - \tan \alpha)(\cos \alpha - \cot \alpha)\). Use the distributive property of multiplication over subtraction:\((\sin \alpha - \tan \alpha)(\cos \alpha - \cot \alpha) = \sin \alpha \cos \alpha - \sin \alpha \cot \alpha - \tan \alpha \cos \alpha + \tan \alpha \cot \alpha\).Now, remember that \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\) and \(\cot \alpha = \frac{\cos \alpha}{\sin \alpha}\). Substitute these definitions into the LHS.

Step 2: Substitute Trigonometric Identities

Substitute the identities for \(\tan \alpha\) and \(\cot \alpha\):\[\sin \alpha \cos \alpha - \sin \alpha \left(\frac{\cos \alpha}{\sin \alpha}\right) - \left(\frac{\sin \alpha}{\cos \alpha}\right) \cos \alpha + \left(\frac{\sin \alpha}{\cos \alpha}\right)\left(\frac{\cos \alpha}{\sin \alpha}\right).\] This simplifies to:\[\sin \alpha \cos \alpha - \cos \alpha - \sin \alpha + 1.\]

Step 3: Simplify the Expression

Now, simplify the expanded expression:\[\sin \alpha \cos \alpha - \cos \alpha - \sin \alpha + 1.\]Rearranging the terms gives:\[(\sin \alpha - 1)(\cos \alpha - 1).\]

Step 4: Verify the Identity with Right Hand Side

Compare the simplified expression from Step 3 to the right-hand side (RHS) of the original identity:\[(\cos \alpha - 1)(\sin \alpha - 1) = (\sin \alpha - 1)(\cos \alpha - 1).\] The LHS simplifies to exactly the same expression as the RHS after rearranging terms or noticing the commutative property, thus verifying the identity.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics that allows us to multiply a single term across terms inside parentheses. It is expressed in the form:

• For addition: \[ a(b + c) = ab + ac \]
• For subtraction: \[ a(b - c) = ab - ac \]

In the context of our exercise, this property was used to expand the expression \[ ( ext{Sin } \alpha - ext{Tan } \alpha)( ext{Cos } \alpha - ext{Cot } \alpha) \]on the left-hand side of the equation. Each term inside the first set of parentheses is multiplied by each term inside the second set, resulting in four products. To understand this clearly:

• Take each part of the first term: \( \text{Sin } \alpha \) and \( \text{Tan } \alpha \)
• Multiply them by each element in the second term: \( \text{Cos } \alpha \) and \( \text{Cot } \alpha \)
• You get: \( \text{Sin } \alpha \cdot \text{Cos } \alpha - \text{Sin } \alpha \cdot \text{Cot } \alpha - \text{Tan } \alpha \cdot \text{Cos } \alpha + \text{Tan } \alpha \cdot \text{Cot } \alpha \)

This efficient method helps break down complex algebraic expressions into more manageable terms, as demonstrated in simplifying the given identity. It's all about distributing multiplications across additions or subtractions!

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, especially when dealing with angles and triangles. In our exercise, we specifically dealt with sine (\(\sin \alpha\)), cosine (\(\cos \alpha\)), tangent (\(\tan \alpha\)), and cotangent (\(\cot \alpha\)). Each serves a unique role and can be defined as:

• \(\sin \alpha\) – Represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• \(\cos \alpha\) – Represents the ratio of the adjacent side to the hypotenuse.
• \(\tan \alpha\) – Defined as the ratio of sine to cosine, or \(\frac{\sin \alpha}{\cos \alpha}\).
• \(\cot \alpha\) – The reciprocal of tangent, or \(\frac{\cos \alpha}{\sin \alpha}\).

Substituting definitions is key when simplifying or transforming equations involving these functions. Often, identities among these functions (like \(\tan \alpha\) and \(\cot \alpha\) in our exercise) help in reducing complex expressions to simpler forms. This step is crucial in verifying the trigonometric identity we worked through. Understanding how these functions interrelate can make complex identities much easier to handle.

Commutative Property

The commutative property tells us that the order of numbers in addition or multiplication does not change the result. For instance:

• Addition: \( a + b = b + a \)
• Multiplication: \( ab = ba \)

In our trigonometric identity problem, the commutative property was highlighted in the final verification step where we simplified the expanded left-hand side to match the structure of the right-hand side. For the final expression \((\sin \alpha - 1)(\cos \alpha - 1)\), it aligns perfectly with \((\cos \alpha - 1)(\sin \alpha - 1)\) due to this property. Recognizing that these products are indeed equal regardless of their order reassures us that we have verified the identity correctly.
It's not just in algebra but also in different areas of mathematics where recognizing and applying the commutative property provides flexibility and ease during calculations. Such properties remind us that there can be multiple correct ways to present or simplify expressions!