Step 1: Express Trigonometric Functions in Sine and Cosine

The first step is to rewrite the trigonometric functions \(\cot\theta\) and \(\csc\theta\) in terms of \(\sin\theta\) and \(\cos\theta\). This helps in algebraic manipulation.

• Recall the definitions: \(\cot\theta = \frac{\cos\theta}{\sin\theta}\)
• Recall the definitions: \(\csc\theta = \frac{1}{\sin\theta}\)

Now, substitute these into the left side of the given equation, \((\cot\theta - \csc\theta)^2\):

\((\cot\theta - \csc\theta)^2 = \left(\frac{\cos\theta}{\sin\theta} - \frac{1}{\sin\theta}\right)^2\)

Step 2: Simplify the Expression Inside the Parentheses

Combine the terms within the parentheses since they have a common denominator (\(\sin\theta\)):

\(\left(\frac{\cos\theta - 1}{\sin\theta}\right)^2\)

Next, square the numerator and the denominator separately:

\(\frac{(\cos\theta - 1)^2}{(\sin\theta)^2} = \frac{(\cos\theta - 1)^2}{\sin^2\theta}\)

Step 3: Apply the Pythagorean Identity

Use the fundamental Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\). We can rearrange this to express \(\sin^2\theta\) in terms of \(\cos\theta\):

\(\sin^2\theta = 1 - \cos^2\theta\)

Substitute this into the denominator of our expression:

\(\frac{(\cos\theta - 1)^2}{1 - \cos^2\theta}\)

Step 4: Factor the Denominator and Simplify

The denominator \(1 - \cos^2\theta\) is a difference of squares, which can be factored as \((1 - \cos\theta)(1 + \cos\theta)\).

The numerator \((\cos\theta - 1)^2\) can be written as \((-(1 - \cos\theta))^2 = (-1)^2(1 - \cos\theta)^2 = (1 - \cos\theta)^2\).

Now the expression becomes:

\(\frac{(1 - \cos\theta)^2}{(1 - \cos\theta)(1 + \cos\theta)}\)

Since \(\theta\) is an acute angle, \(0 < \theta < \frac{\pi}{2}\), which means \(\cos\theta \neq 1\) and \(\sin\theta \neq 0\). Therefore, \(1 - \cos\theta \neq 0\), and we can cancel one factor of \((1 - \cos\theta)\) from the numerator and the denominator:

\(\frac{1 - \cos\theta}{1 + \cos\theta}\)

Step 5: Determine the Denominator D

We are given the equation:

\((\cot\theta - \csc\theta)^2 = \frac{1 - \cos\theta}{D}\)

From our simplification in Step 4, we found:

\((\cot\theta - \csc\theta)^2 = \frac{1 - \cos\theta}{1 + \cos\theta}\)

By comparing the two forms, we can equate the denominators:

\(D = 1 + \cos\theta\)

Thus, the denominator D is \(1 + \cos\theta\).