Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following.Consider the expression \(\tan \left(\frac{\pi}{2}+x\right)\) (a) Why can't we use the identity for \(\tan (A+B)\) to express it as a function of \(x\) alone? (b) Use the identity \(\tan \theta=\frac{\sin \theta}{\cos \theta}\) to rewrite the expression in terms of sine and cosine. (c) Use the result of part (b) to show that $$ \tan \left(\frac{\pi}{2}+x\right)=-\cot x $$

Step-by-Step Solution

Answer

(a) The identity involves \( \tan \left(\frac{\pi}{2} \right)\), which is undefined. (b) Express as \( \frac{\cos x}{-\sin x}\). (c) Simplifies to \(-\cot x\).

Step 1: Understanding Limitations of Sum Angle Identity

The sum identity for tangent, \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \), involves calculating \( \tan \left(\frac{\pi}{2}\right) \). However, the tangent of \( \frac{\pi}{2} \) is undefined, creating an issue when trying to rewrite \( \tan \left(\frac{\pi}{2} + x\right) \) solely using this identity.

Step 2: Rewrite Using Sine and Cosine

Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can express \( \tan \left(\frac{\pi}{2} + x\right) \) as:\[\tan \left(\frac{\pi}{2} + x\right) = \frac{\sin\left(\frac{\pi}{2} + x\right)}{\cos\left(\frac{\pi}{2} + x\right)}.\] The addition formulas for sine and cosine are:\[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta,\]\[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta.\] Therefore,\[\sin\left(\frac{\pi}{2} + x\right) = \cos x, \quad \cos\left(\frac{\pi}{2} + x\right) = -\sin x.\]So, \( \tan \left(\frac{\pi}{2} + x\right) = \frac{\cos x}{-\sin x} \).

Step 3: Simplify to the Desired Result

The expression \( \frac{\cos x}{-\sin x} \) simplifies to \(-\frac{\cos x}{\sin x} \), which is equivalent to \(-\cot x \) because \( \cot x = \frac{\cos x}{\sin x} \). Thus, \[\tan \left(\frac{\pi}{2}+x\right)=-\cot x.\]

Key Concepts

Tangent sum identity

The tangent sum identity is a handy formula in trigonometry used to expand expressions like \( \tan(A + B) \). It states: \[\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\,\]which allows us to express the tangent of a sum of two angles in terms of their individual tangents. But there's a catch when using this identity with \( \frac{\pi}{2} \) because the tangent of \( \frac{\pi}{2} \) is undefined. Here's why:

• As we know, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• When \( \theta = \frac{\pi}{2} \), \( \cos \theta = 0 \), making \( \tan \theta \) undefined due to division by zero.
• Thus, trying to use the sum identity here doesn't work as intended.

This limitation is crucial because it directs us to consider other identities or properties when handling such cases.

Sine and cosine identities

To express \( \tan \left(\frac{\pi}{2} + x\right) \) using sine and cosine, we utilize the identity: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]. This helps us navigate around the issue with \( \tan \frac{\pi}{2} \) being undefined. Here's how it's done:

• We start with the addition formulas:
  • \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
  • \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)
• Substituting \( \alpha = \frac{\pi}{2} \) and \( \beta = x \):
  • \( \sin\left(\frac{\pi}{2} + x\right) = \cos x \)
  • \( \cos\left(\frac{\pi}{2} + x\right) = -\sin x \)

Therefore, \( \tan \left(\frac{\pi}{2} + x\right) = \frac{\cos x}{-\sin x} \). This identity is more versatile for such angle manipulations.

Cotrangent

The cotangent function, commonly denoted as \( \cot \theta \), is another fundamental concept in trigonometry. It is defined as the reciprocal of the tangent function: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \].This relationship comes in handy when simplifying expressions involving angles.Here's why \( \cot x \) is essential in this context:

• We have expressed \( \tan \left(\frac{\pi}{2} + x\right) \) as \( \frac{\cos x}{-\sin x} \).
• Notice this is simply \( -\frac{\cos x}{\sin x} \), which looks similar to \( -\cot x \).
• Thus, the expression further simplifies to \( -\cot x \).

Understanding \( \cot x \) in this way allows us to make these connections and simplifications, showcasing the beauty and harmony of trigonometric identities.