Problem 31 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$(\csc t-\cot t)^{4}(\csc t+\cot t)^{4}=1$$

Step-by-Step Solution

Answer

The identity is verified as true by simplification.

Step 1: Understand the Identity

The identity we need to verify is \((\csc t - \cot t)^{4}(\csc t + \cot t)^{4} = 1\). This means we need to demonstrate that when we expand the left-hand side, it equals 1.

Step 2: Use Difference of Squares

Notice that \((a - b)(a + b) = a^2 - b^2\). Here, let \(a = \csc t\) and \(b = \cot t\). We use this formula to rewrite the expression as \(((\csc t)^2 - (\cot t)^2)^2\).

Step 3: Use Trigonometric Identities

Recall that \(\csc t = \frac{1}{\sin t}\) and \(\cot t = \frac{\cos t}{\sin t}\). Therefore, \((\csc t)^2 = \frac{1}{\sin^2 t}\) and \((\cot t)^2 = \frac{\cos^2 t}{\sin^2 t}\). Substitute these expressions into the equation to get \[\left(\frac{1}{\sin^2 t} - \frac{\cos^2 t}{\sin^2 t}\right)^2.\]

Step 4: Simplify the Expression

Combine the terms inside the parentheses: \[\frac{1 - \cos^2 t}{\sin^2 t} = \frac{\sin^2 t}{\sin^2 t} = 1.\]The expression thus simplifies to \((1)^2 = 1\).

Step 5: Verify the Identity

Since the simplified version of the expression is 1, the original identity \((\csc t - \cot t)^4(\csc t + \cot t)^4 = 1\) is true.

Key Concepts

Understanding Cosecant

Cosecant is an important trigonometric function, often abbreviated as \( \csc \). It is the reciprocal of the sine function. If you think of it in terms of a right triangle, the cosecant of an angle \( t \) is the length of the hypotenuse divided by the length of the opposite side: \[ \csc t = \frac{1}{\sin t} = \frac{\text{hypotenuse}}{\text{opposite side}} \] This can be particularly useful when you have the length of the hypotenuse and you need the opposite side or vice versa.

• Trigonometric functions are often used in equations and identities in math. They are essential for solving problems in physics and engineering.
• Cosecant is less commonly used but is crucial in certain contexts, especially when dealing with angles that aren't right angles.

Remember that as the sine value gets smaller, the cosecant value becomes larger. That's because dividing by a smaller number (a denominator close to zero) gives a larger result.

Exploring Cotangent

Cotangent is a trigonometric function represented as \( \cot \) and is the reciprocal of the tangent function. It can be defined using a right triangle: it is the adjacent side over the opposite side. The equation for cotangent is: \[ \cot t = \frac{1}{\tan t} = \frac{\cos t}{\sin t} = \frac{\text{adjacent}}{\text{opposite}} \] Cotangent is helpful in situations where you know the lengths of the adjacent and opposite sides and need to find the relationship of the angle with these sides.

• This function is less intuitive than sine or cosine but equally useful.
• It’s particularly important when working with angles in the context of periodic functions and waves.

Understanding cotangent is crucial for solving complex trigonometric identities, like the one in the exercise above.

Utilizing the Difference of Squares

The difference of squares is a widely-used algebraic formula represented by the identity \( (a-b)(a+b) = a^2 - b^2 \). It’s a vital concept when simplifying complex algebraic expressions. This formula works well because multiplying \( a-b \) by \( a+b \) cancels out the middle terms, leaving you with just the squares of each term.

• Look at the form: \( a^2 - b^2 \). It's worth recognizing squares in an expression whenever possible.
• This concept is not limited to numbers; it effectively applies to variables, including trigonometric functions.

In the given exercise, applying the difference of squares simplifies the identity significantly, setting up a clear path to solving the problem.