Problem 56 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{\cos (-t)}{1+\tan (-t)}-\frac{\sin (-t)}{1+\cot (-t)}=\sin t+\cos t $$

Step-by-Step Solution

Answer

The identity is verified as the expression simplifies to \( \sin t + \cos t \).

Step 1: Recall Trigonometric Identities

First, recall that \( \cos(-t) = \cos t \), \( \sin(-t) = -\sin t \), \( \tan(-t) = -\tan t \), and \( \cot(-t) = -\cot t \). We'll use these identities to simplify both sides of the equation.

Step 2: Rewrite the Expression

Rewrite \( \frac{\cos(-t)}{1+\tan(-t)} - \frac{\sin(-t)}{1+\cot(-t)} \) using the identities: \( \frac{\cos t}{1-\tan t} - \frac{-\sin t}{1-\cot t} \). Simplifying further gives us \( \frac{\cos t}{1-\tan t} + \frac{\sin t}{1-\cot t} \).

Step 3: Rationalize the Denominators

Multiply numerator and denominator of each fraction by the conjugate of their respective denominators: for \( \frac{\cos t}{1-\tan t} \), multiply by \( \frac{1+\tan t}{1+\tan t} \), and for \( \frac{\sin t}{1-\cot t} \), multiply by \( \frac{1+\cot t}{1+\cot t} \).

Step 4: Simplify the Expressions

Simplifying gives:\[ \frac{\cos t (1+\tan t)}{1 - \tan^2 t} + \frac{\sin t (1+\cot t)}{1 - \cot^2 t} \]Using \( 1 - \tan^2 t = \frac{\cos^2 t - \sin^2 t}{\cos^2 t} \) and \( 1 - \cot^2 t = \frac{\sin^2 t - \cos^2 t}{\sin^2 t} \), we simplify further.

Step 5: Combine Like Terms

Combine and further simplify the rationalized expressions. Set common denominators to combine:\[ \frac{\cos^2 t (1+\tan t) \sin^2 t + \sin^2 t (1+\cot t) \cos^2 t}{\cos^2 t \sin^2 t} \]

Step 6: Verify and Simplify the Combined Expression

Expand both numerators and simplify. Use the identity \( \sin^2 t + \cos^2 t = 1 \) to simplify the expression into \( \sin t + \cos t \), matching the right-hand side.

Key Concepts

Cosine Identity

The cosine function has a fascinating property when taking the negative of an angle. Specifically, the cosine identity states that \( \cos(-t) = \cos t \). This property is because cosine is an even function. In simple terms, even functions mirror themselves across the vertical axis, meaning that the cosine graph looks the same whether you consider a positive or negative angle.
Understanding even functions is vital when verifying identities because it allows the substitution of \( \cos(-t) \) with \( \cos t \) effortlessly. This principle simplifies expressions by making calculations consistent, regardless of the angle's sign. Overall, recognizing and applying the cosine identity reduces complexity in trigonometric problems.

Sine Identity

Sine, unlike cosine, is an odd function. This means that the sine function has a unique property: \( \sin(-t) = -\sin t \). This property is intuitive if you visualize the sine wave, which flips across the horizontal axis when considering negative angles.
The odd nature of the sine function means that the sine of a negative angle is simply the negative of the sine of the positive angle. In trigonometric identities, this understanding allows us to transform expressions involving negative sine values smoothly into their positive counterparts, helping greatly when simplifying or verifying identities. By switching \( \sin(-t) \) to \(-\sin t \), calculations become more straightforward, adhering to simpler arithmetic rules.

Tangent and Cotangent Identities

Tangent and cotangent are reciprocals of one another, and their interactions are much like sine and cosine. The tangent identity for negative angles is \( \tan(-t) = -\tan t \), and similarly, \( \cot(-t) = -\cot t \). Like sine, both tangent and cotangent are odd functions, meaning they also reflect across the horizontal axis for negative angles.
Understanding these identities helps when dealing with trigonometric expressions that involve tangent and cotangent. By recognizing that you can replace \( \tan(-t) \) and \( \cot(-t) \) with \(-\tan t \) and \(-\cot t \) respectively, you simplify expressions and establish a pattern. This reduction aids when multiplying or dividing terms in an expression, leading to more manageable calculations.
In practice, these identities are applied frequently when rationalizing denominators, as they help in aligning expression forms with those that lead directly to identity verification.

Simplifying Expressions

Simplifying trigonometric expressions often involves using identities to re-write parts of expressions in a more manageable way. The goal is to reduce complexity and make the expressions easier to compare or solve. Key to this is recognizing which identities apply and using algebraic techniques like multiplying by conjugates or combining like terms.
In trigonometry, one common technique is rationalizing the denominator. This means multiplying both the numerator and denominator by the conjugate of the denominator. For example, the expression \( \frac{\cos t}{1 - \tan t} \) can be simplified by multiplying by \( \frac{1 + \tan t}{1 + \tan t} \). This technique helps eliminate complex terms and creates more uniform denominators.

• Recognize and apply pertinent identities: Such as \( \sin^2 t + \cos^2 t = 1 \).
• Multiply by conjugates: Simplifies expressions by eliminating radicals or complex fractions.
• Combine like terms: Makes it possible to consolidate expressions for simpler results.

Ultimately, simplifying expressions using these methods transforms intricate trigonometric problems into simpler, solvable equations, paving the way for straightforward identity verification.