Problem 143 - Calculus Volume 1

Question

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. $$(1+\tan \theta)^{2}-2 \tan \theta$$

Step-by-Step Solution

Answer

\((1+\tan \theta)^{2} - 2 \tan \theta = \sec^2 \theta\).

Step 1: Expand the Square

Let's begin by expanding the squared term in the expression \((1+\tan \theta)^{2}\). This follows the algebraic identity \((a+b)^{2} = a^{2} + 2ab + b^{2}\). Here, \(a=1\) and \(b=\tan \theta\). Expanding, we get:\[(1+\tan \theta)^{2} = 1^{2} + 2 \cdot 1 \cdot \tan \theta + (\tan \theta)^{2} = 1 + 2\tan \theta + \tan^2 \theta\]

Step 2: Rewrite with Basic Identities

We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Using this, replace \(\tan \theta\) and \(\tan^2 \theta\) in the expanded expression:\[1 + 2\frac{\sin \theta}{\cos \theta} + \left(\frac{\sin \theta}{\cos \theta}\right)^2\].

Step 3: Handle the \(\tan^2\) Term

Observe that \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\). Substitute this back into the expression:\[1 + 2 \frac{\sin \theta}{\cos \theta} + \frac{\sin^2 \theta}{\cos^2 \theta}\].

Step 4: Simplify the Expression Further

Combine the terms in the expression. Start with the expression we derived in terms of sine and cosine:\[1 + 2 \frac{\sin \theta}{\cos \theta} + \frac{\sin^2 \theta}{\cos^2 \theta} - 2 \frac{\sin \theta}{\cos \theta}\].Notice that the \(2 \frac{\sin \theta}{\cos \theta}\) terms cancel each other out.

Step 5: Simplified Result

After canceling, what remains is:\[1 + \frac{\sin^2 \theta}{\cos^2 \theta} = 1+ \tan^2 \theta.\]This is a common trigonometric identity, equal to \(\sec^2 \theta\). Therefore:\[(1+\tan \theta)^{2} - 2 \tan \theta = \sec^2 \theta\].

Key Concepts

Simplifying Trigonometric Expressions

To simplify trigonometric expressions, we often begin by rewriting the terms in their basic trigonometric forms. One approach is to express tangent, cosecant, secant, and cotangent in terms of sine and cosine.
This helps us standardize the expression. For instance:

• Tangent, \( \tan \theta \), is expressed as \( \frac{\sin \theta}{\cos \theta} \).
• Similarly, \( \tan^2 \theta \) can be transformed to \( \frac{\sin^2 \theta}{\cos^2 \theta} \).

By converting complex trigonometric functions into basic sine and cosine terms, the expression becomes easier to manage.
This technique facilitates the application of further algebraic operations, such as expansion, factorization, or cancellation of terms.

Sine and Cosine Transformation

Transforming trigonometric expressions using sine and cosine can simplify complex problems and reveal underlying identities. For example, when working with \( (1 + \tan \theta)^2 \), converting \( \tan \theta \) into its sine and cosine components allows us to use fundamental trig identities.
The identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is pivotal here.
Benefits:

• Allows the expression to be broken down into more familiar parts.
• Helps identify opportunities for simplification.

Once transformed, standard algebraic techniques such as expansion or reduction can be applied more easily. This not only simplifies calculations but can also lead to recognizing classic trigonometric identities, making the expression more elegant and tractable.

Algebraic Identities in Trigonometry

Algebraic identities serve as powerful tools in solving trigonometric problems. The problem at hand leverages the identity \( (a + b)^2 = a^2 + 2ab + b^2 \) to expand \((1 + \tan \theta)^2\).
This step reveals the distinct terms that might be simplified further. Beyond basic algebra, trigonometric identities form a bridge that connects different trigonometric functions.
Key Identities:

• The Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
• The identity \( 1 + \tan^2 \theta = \sec^2 \theta \).

Recognizing and applying these identities simplifies the expression, as in this exercise, linking \(1 + \tan^2 \theta\) to \(\sec^2 \theta\). These connections are crucial for solving trigonometric equations and transforming expressions to more usable or insightful forms.