Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\sin ^{2} \beta\left(1+\cot ^{2} \beta\right)=1\)

Step-by-Step Solution

Answer

The equation \( \sin^2 \beta (1 + \cot^2 \beta) = 1 \) is an identity.

Step 1: Understand the Identity

The given expression is \( \sin^2 \beta (1 + \cot^2 \beta) \). Our goal is to show that it equals 1. Identities in trigonometry are equations that hold true for all values of the variable where both sides are defined.

Step 2: Recall Trigonometric Identities

We need to recall the trigonometric identity for \( 1 + \cot^2 \theta \), which is \( \csc^2 \theta \). This can be derived from the identity \( \csc^2 \theta = 1 + \cot^2 \theta \).

Step 3: Substitute \(1 + \cot^2 \beta\)

Using the identity \( 1 + \cot^2 \beta = \csc^2 \beta \), substitute \( \csc^2 \beta \) into the equation so it becomes \( \sin^2 \beta \times \csc^2 \beta \).

Step 4: Simplify Using \( \csc^2 \beta\) Definition

Recall that \( \csc \beta = \frac{1}{\sin \beta} \), hence \( \csc^2 \beta = \frac{1}{\sin^2 \beta} \). Substitute this in: \( \sin^2 \beta \times \frac{1}{\sin^2 \beta} \).

Step 5: Simplify the Expression

On simplifying \( \sin^2 \beta \times \frac{1}{\sin^2 \beta} = 1 \) since the \( \sin^2 \beta \) terms cancel each other out.

Step 6: Conclusion

The original expression simplifies to 1, confirming it is an identity.

Key Concepts

Sine Function

The sine function, denoted as \( \sin \theta \), is one of the fundamental trigonometric functions. It relates the angle in a right-angle triangle to the ratio of the length of the opposite side to the hypotenuse. In mathematical terms, it is expressed as:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

Sine is essential for understanding wave motion and oscillations. It's periodic, with a repeating cycle every \( 360^\circ \) or \( 2\pi \) radians. The function oscillates between -1 and 1, making it valuable for modeling scenarios in physics and engineering. In our exercise, \( \sin^2 \beta \) is used, which means \( (\sin \beta)^2 \). This square plays a key role in simplifying expressions involving other trigonometric identities.

Cosecant Function

The cosecant function is the reciprocal of the sine function. It is denoted as \( \csc \theta \) and is defined as the ratio of the hypotenuse to the opposite side:

• \( \csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}} \)

This function is less commonly discussed than sine, cosine, or tangent but is crucial in scenarios where reciprocal identities are advantageous. In the solution, \( \csc^2 \beta \) was derived using a Pythagorean identity, which states \( 1 + \cot^2 \beta = \csc^2 \beta \). This helps with the simplification process, which turns a complex expression into a simpler equation.

Cotangent Function

The cotangent function is another reciprocal trigonometric function. It is the reciprocal of the tangent function and is denoted by \( \cot \theta \). It is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} = \frac{\text{adjacent}}{\text{opposite}} \)

Understanding \( \cot \theta \) is vital for dealing with certain trigonometric identities requiring simplification or transformation. In the exercise, it is used within an identity that relates it to the cosecant function, playing a significant role in verifying that the given trigonometric equation is an identity.

Simplifying Expressions

Simplifying expressions involving trigonometric functions often makes complex equations easier to understand and solve. This involves using known identities to rewrite expressions in a simpler or more recognizable form. Here are some tips for simplifying:

• Use basic trigonometric identities like \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Apply reciprocal identities such as \( \csc \theta = \frac{1}{\sin \theta} \) whenever needed.
• Look for opportunities to cancel terms, as with \( \sin^2 \beta \times \frac{1}{\sin^2 \beta} = 1 \) in this exercise.

Words like "simplify" often mean performing these operations to reach an easier-to-manage form of the original equation. It's like tidying up an equation so its true value or relationship emerges clearly. This process is what allowed the identity \( \sin^2 \beta (1 + \cot^2 \beta) = 1 \) to be verified.