Problem 143 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(\sec^2 \theta\).

Step 1: Expand the Expression

Start with the given expression \((1+\tan \theta)^{2}-2 \tan \theta\). First, expand \((1+\tan \theta)^{2}\) using the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\). This gives us:\[(1 + \tan \theta)^2 = 1^2 + 2(1)(\tan \theta) + (\tan \theta)^2 = 1 + 2\tan \theta + \tan^2 \theta\]

Step 2: Substitute Expanded Form Back

Substitute the expanded form back into the original expression:\[1 + 2\tan \theta + \tan^2 \theta - 2 \tan \theta\]

Step 3: Simplify the Expression

Now, simplify the expression by combining like terms:The terms \(+2\tan \theta\) and \(-2\tan \theta\) cancel each other out, leaving us with:\[1 + \tan^2 \theta\]

Step 4: Use Trigonometric Identity

Recognize that \(1 + \tan^2 \theta\) is a known trigonometric identity, which equals \(\sec^2 \theta\). Thus, substitute to simplify:\[\sec^2 \theta\]

Key Concepts

Sine and Cosine Functions

To simplify trigonometric expressions, it's helpful to express them in terms of sine and cosine. The tangent function, for instance, can be expressed as the ratio of sine to cosine:

• Tan theta as \[\tan \theta = \frac{\sin \theta}{\cos \theta}\]This transformation is crucial when simplifying, as it allows you to work with basic sine and cosine, which are more straightforward.

Additionally, other trigonometric identities are also often expressed in terms of sine and cosine for the same reason. When dealing with an expression involving more complex trigonometric functions, rewriting them using sine and cosine can make it easier to understand and simplify the mathematics involved. Breaking down complex expressions into these primary functions is a building block technique in trigonometry, giving a clearer path to apply identities or algebraic manipulation.

Algebraic Expressions

Algebraic expressions in trigonometry often include identities like \((a+b)^2 = a^2 + 2ab + b^2\). For the given problem, we expanded \((1 + \tan \theta)^2\) using this identity to assist in simplification. This step is crucial because combining like terms or eliminating redundant terms can significantly reduce the complexity of an expression. It's about seeing patterns and using known formulas to reconfigure the expression for easier manipulation.

• Expanding Expressions: This involves applying algebraic identities to break down the expression into simpler parts or terms.
• Combining Like Terms: Once expanded, look for numbers or variables that can cancel each other out, such as \(+2\tan \theta\) and \(-2\tan \theta\) in this case.

By applying these techniques, students can simplify complications in expressions without needing to directly solve complex equations.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves recognizing well-known identities and using them to our advantage. In the example, we identify \(1 + \tan^2 \theta\) as part of a standard identity, specifically \(\sec^2 \theta = 1 + \tan^2 \theta\). By substituting this identity back into the expression, we reduce it from something involving two terms into a single, more manageable term.This process is vital because:

• Efficient Calculations: Using identities can transform complex expressions into simpler ones, making calculations more straightforward and less time-consuming.
• Clear Understanding: Recognizing these identities helps reinforce a deeper understanding of how different trigonometric functions relate to one another.

By familiarizing themselves with these identities, students can significantly improve their ability to simplify even the most complicated trigonometric expressions. Mastery of identities and the ability to apply them is an essential skill in trigonometry.