Problem 90 - Precalculus Mathematics for Calculus

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 to be true; both sides simplify to \((\cos \alpha - 1)(\sin \alpha - 1)\).

Step 1: Start with the Left Side

Let's start by taking the expression on the left side of the equation \((\sin \alpha - \tan \alpha)(\cos \alpha - \cot \alpha)\). First, rewrite \(\tan \alpha\) and \(\cot \alpha\) in terms of sine and cosine.- \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\)- \(\cot \alpha = \frac{\cos \alpha}{\sin \alpha}\)Substituting these into the expression gives:\[(\sin \alpha - \frac{\sin \alpha}{\cos \alpha})(\cos \alpha - \frac{\cos \alpha}{\sin \alpha})\]

Step 2: Simplify the Expression

For each factor, factor out common terms:1. For \(\sin \alpha - \frac{\sin \alpha}{\cos \alpha}\) Factor out \(\sin \alpha\): \[ \sin \alpha (1 - \frac{1}{\cos \alpha}) \] Which simplifies to: \[ \sin \alpha \left(\frac{\cos \alpha - 1}{\cos \alpha}\right) \]2. For \(\cos \alpha - \frac{\cos \alpha}{\sin \alpha}\) Factor out \(\cos \alpha\): \[ \cos \alpha (1 - \frac{1}{\sin \alpha}) \] Which simplifies to: \[ \cos \alpha \left(\frac{\sin \alpha - 1}{\sin \alpha}\right) \]

Step 3: Combine the Simplified Factors

Multiply the two simplified factors:\[\left(\sin \alpha \frac{\cos \alpha - 1}{\cos \alpha}\right) \times \left(\cos \alpha \frac{\sin \alpha - 1}{\sin \alpha}\right)\]Simplifying, we have:\[\frac{(\sin \alpha)(\cos \alpha - 1)(\cos \alpha)(\sin \alpha - 1)}{\cos \alpha \sin \alpha}\]Cancel out \(\sin \alpha\) and \(\cos \alpha\) in the fraction:\[(\cos \alpha - 1)(\sin \alpha - 1)\]

Step 4: Compare with the Right Side

The expression simplified in Step 3, \((\cos \alpha - 1)(\sin \alpha - 1)\), is exactly the right-hand side of the identity to verify.Thus, the identity \[(\sin \alpha - \tan \alpha)(\cos \alpha - \cot \alpha) = (\cos \alpha - 1)(\sin \alpha - 1)\]is verified to be true.

Key Concepts

Sine

The sine function, commonly denoted as \( \sin \), is a fundamental concept in trigonometry. It helps in understanding and analyzing right-angled triangles. In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The basic formula is:\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]Sine values vary depending on the angle \( \alpha \), ranging from -1 to 1. Here are some key angles and their sine values:

• For \(0^{\circ}\) and \(180^{\circ}\), \( \sin \alpha = 0 \).
• For \(90^{\circ}\), \( \sin \alpha = 1 \).
• For \(270^{\circ}\), \( \sin \alpha = -1 \).

The sine function is also periodic, repeating its values every 360°, or \(2\pi\) radians. In trigonometric identities like the one given in this exercise, sine is often expressed using other functions to help simplify the expressions or verify identities. In our problem, we rearrange terms involving sine to facilitate simplification.

Cosine

The cosine function, noted as \( \cos \), plays a crucial role in trigonometry, hand in hand with sine. In a right-angled triangle, the cosine of an angle measures the ratio of the adjacent side to the hypotenuse. The formula is:\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]Cosine values range similarly to sine, from -1 to 1. Important angle relationships are:

• \( \cos 0^{\circ} = 1 \) and \( \cos 180^{\circ} = -1 \).
• \( \cos 90^{\circ} = 0 \) and \( \cos 270^{\circ} = 0 \).

Like sine, cosine is also periodic, and it repeats every 360°, or \(2\pi\) radians. Cosine is often used in conjunction with sine to simplify expressions in identities using the Pythagorean identity: \[ \sin^2 \alpha + \cos^2 \alpha = 1 \]In the given exercise, cosine functions are manipulated to verify the identity, proving versatility in formulating and rearranging terms to achieve simplification.

Tangent

Tangent, represented as \( \tan \), extends the relationship between the sine and cosine functions. It is defined as the ratio of the sine of an angle to its cosine:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]This definition makes tangent valuable in various trigonometry problems, especially those involving slopes and angles. Key angles for tangent include:

• \( \tan 0^{\circ} = 0 \)
• \( \tan 45^{\circ} = 1 \)
• \( \tan 90^{\circ} \) is undefined, as \( \cos 90^{\circ} = 0 \)

Tangent values are valid and repeating every 180° or \(\pi\) radians. It becomes undefined at 90° and 270°, where cosine is zero. In our problem, tangent was rewritten in terms of sine and cosine to verify the identity: \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \).This transformation allowed us to meet the other side of the identity efficiently, showing the critical role of tangent in simplifying trigonometric relationships.