Question

Simplify the following expressions by writing each one using a single trigonometric function. $$ 9 \sec ^{2} \theta-9 $$

Short answer

The expression simplifies to \( 9 \tan^2 \theta \).

Step-by-step solution

Recognize the identity

We know one of the Pythagorean identities that relates cosine and secant: \[ \sec^2 \theta = 1 + \tan^2 \theta \] This identity will help us to simplify the given expression.

Substitute using identity

Substitute \( \sec^2 \theta \) in the expression with \( 1 + \tan^2 \theta \). This gives us:\[ 9 (1 + \tan^2 \theta) - 9 \]

Distribute and simplify

Distribute the 9 into the expression inside the parenthesis:\[ 9 + 9 \tan^2 \theta - 9 \]Next, simplify by combining the constant terms (9 and -9):\[ 9 \tan^2 \theta \]

Final simplification

The expression simplifies to:\[ 9 \tan^2 \theta \] This is a single trigonometric function, as required.

Key concepts

Pythagorean identities

In trigonometry, the Pythagorean identities are a set of fundamental relationships between the three basic trigonometric functions: sine, cosine, and tangent. They provide a bridge for simplifying and transforming expressions involving trigonometric functions. One of the key Pythagorean identities is:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity can be manipulated to derive two other important forms:

• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

These equations are derived by dividing the original identity by either \(\cos^2 \theta\) or \(\sin^2 \theta\). In the context of the problem we are solving, the identity \(\sec^2 \theta = 1 + \tan^2 \theta\) is particularly useful. It allows us to replace \(\sec^2 \theta\) in expressions, which simplifies calculations and unlocks a way to transform complex trigonometric equations into simpler forms. Understanding these identities is crucial for effectively navigating various trigonometric problems.

trigonometric simplification

Trigonometric simplification involves rewriting complex trigonometric expressions in simpler or more useful forms. This frequently employs trigonometric identities like the Pythagorean identities to reduce expressions to a single trigonometric function or a more convenient form for computation. For example, the expression \( 9\sec^2\theta - 9 \) may initially seem complex, but using the identity \(\sec^2\theta = 1 + \tan^2 \theta\), it can be simplified effectively.Here’s the simplification step-by-step:

• Recognize the Pythagorean identity: \(\sec^2\theta = 1 + \tan^2\theta\).
• Substitute \(\sec^2\theta\) with \(1 + \tan^2\theta\), giving you \(9(1 + \tan^2\theta) - 9\).
• Distribute the 9 inside the parentheses: \(9 + 9\tan^2\theta - 9\).
• Simplify by cancelling the \(9\) and \(-9\), leaving you with \(9\tan^2\theta\).

This process illustrates how complex expressions can be distilled into simpler forms, making them easier to work with and understand.

secant function

The secant function, denoted as \(\sec\theta\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function:\[ \sec\theta = \frac{1}{\cos\theta} \]This definition makes it an important function in trigonometry, especially when dealing with expressions involving division by the cosine function. The secant function is most commonly seen in trigonometric identities, like the Pythagorean identity involving secant and tangent given by:\[ \sec^2 \theta = 1 + \tan^2 \theta \]Understanding this identity is key to simplifying expressions that involve \(\sec^2\theta\). By recognizing this identity in trigonometric problems, one can often transform complicated expressions into simpler terms. The secant function peaks at specific points where the cosine values approach zero, which is significant in understanding its behavior and graphical interpretations. Mastery of the secant function and its interplay with other trigonometric functions is essential for anyone delving deeply into trigonometric analyses.