Problem 88 - 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 correct; both sides simplify to \((\cos \alpha - 1)(\sin \alpha - 1)\).

Step 1: Express All Trigonometric Functions in terms of Sine and Cosine

To begin, let's rewrite all the trigonometric functions in terms of \( \sin \alpha \) and \( \cos \alpha \). Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \). Substitute these into the equation:\[ (\sin \alpha - \frac{\sin \alpha}{\cos \alpha})(\cos \alpha - \frac{\cos \alpha}{\sin \alpha}) \]

Step 2: Simplify Each Term of the Expression

Simplify each part of the expression separately:- For \( \sin \alpha - \frac{\sin \alpha}{\cos \alpha} \): \[ \sin \alpha (1 - \frac{1}{\cos \alpha}) = \sin \alpha \cdot \frac{\cos \alpha - 1}{\cos \alpha} \]- For \( \cos \alpha - \frac{\cos \alpha}{\sin \alpha} \): \[ \cos \alpha (1 - \frac{1}{\sin \alpha}) = \cos \alpha \cdot \frac{\sin \alpha - 1}{\sin \alpha} \]

Step 3: Multiply the Simplified Expressions

Now that both expressions are simplified, multiply them together:\[\left( \sin \alpha \cdot \frac{\cos \alpha - 1}{\cos \alpha} \right) \cdot \left( \cos \alpha \cdot \frac{\sin \alpha - 1}{\sin \alpha} \right) = \frac{\sin \alpha \cos \alpha (\cos \alpha - 1)(\sin \alpha - 1)}{\cos \alpha \sin \alpha}\]The terms \( \sin \alpha \) and \( \cos \alpha \) in the numerator and denominator cancel out, leaving:\[ (\cos \alpha - 1)(\sin \alpha - 1) \]

Step 4: Verify the Identity

Both sides of the original equation simplify to \((\cos \alpha - 1)(\sin \alpha - 1)\). This confirms that the identity is correct:\[(\sin \alpha - \tan \alpha)(\cos \alpha - \cot \alpha) = (\cos \alpha - 1)(\sin \alpha - 1)\]Hence, the identity is verified.

Key Concepts

Sine and Cosine Functions

One of the fundamental aspects of trigonometry revolves around the sine and cosine functions. These functions play a critical role as the foundation of many other trigonometric functions. To understand their importance, let's delve into what these functions represent:

• Sine (\(\sin\)) function: Given an angle \(\alpha\) in a right triangle, \(\sin\) of that angle is the ratio of the length of the side opposite the angle to the hypotenuse. Mathematically, it is written as \(\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}\).
• Cosine (\(\cos\)) function: Similarly, the \(\cos\) of an angle is the ratio of the length of the adjacent side to the hypotenuse, expressed as \(\cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}}\).

These definitions provide the basis for other trigonometric functions like tangent (\(\tan\)) and cotangent (\(\cot\)), which can be expressed in terms of \(\sin\) and \(\cos\). For example, \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\) and \(\cot \alpha = \frac{\cos \alpha}{\sin \alpha}\).

Understanding these relationships helps simplify expressions and verify identities, as we'll explore further.

Simplifying Expressions

Simplifying trigonometric expressions often involves reducing complex expressions into simpler forms using known identities and relationships between the functions involved. To simplify effectively, consider these steps:

• Expressing in terms of sine and cosine: Often, converting all terms into sine and cosine functions helps identify common factors or cancellations. This was done in the exercise when rewriting \(\tan \alpha\) and \(\cot \alpha\) in terms of \(\sin \alpha\) and \(\cos \alpha\).


• Finding a common factor: Factors can sometimes be combined and simplified, especially when fraction parts are involved. In the exercise, the expression \(\sin \alpha - \frac{\sin \alpha}{\cos \alpha}\) was simplified by factoring out \(\sin \alpha\).


• Cancelling common terms: When multiplying simplified expressions, look for common terms in the numerator and the denominator that can be cancelled out, making the expression even simpler.

By consistently applying these strategies, we can simplify even complex trigonometric expressions quickly and efficiently.

Verifying Trigonometric Identities

Verifying trigonometric identities involves proving that two expressions are equivalent for all values in their domains. It's a common task in trigonometry that helps deepen understanding of function properties and relationships.

• Start with one side: Begin by transforming one side of the equation, aiming to make it look like the other side. This involves using trigonometric identities and algebraic manipulation.


• Use known identities: Employ standard trigonometric identities, such as Pythagorean identities, to assist in transforming the expressions. For example, in our exercise, the expressions were simplified using basic trigonometric principles.


• Simplify and compare: After applying the identities and simplifying the expression, compare it with the other side of the equation. As seen in the exercise, both sides ended up being \((\cos \alpha - 1)(\sin \alpha - 1)\), verifying the identity.

Understanding and using these steps can make verifying trigonometric identities a straightforward and insightful process.