Question

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$$\begin{tabular}{|l|l|} \\hline statement & rule \\\\ \\hline \\((1 - \\cos x)(1 + \\cos x)\\) & \\\\ \\hline \\(= 1 - \\cos^2 x\\) & rule? \\\\ \\hline \\(= \\sin^2 x\\) & rule? \\\\ \\hline \\(= \\frac{1}{\\csc^2 x}\\) & rule? \\\\ \\hline \\(= \\frac{1}{1 + \\cot^2 x}\\) & rule? \\\\ \\hline \\end{tabular}$$

Explanation:

Identify the algebraic expansion

Using the Trigonometric Identities knowledge point

$$ (1 - \cos x)(1 + \cos x) = 1^2 - \cos^2 x = 1 - \cos^2 x $$

This step uses basic algebraic multiplication (difference of squares).

Apply the Pythagorean identity

Using the Pythagorean Identities knowledge point

$$ \sin^2 x + \cos^2 x = 1 \implies 1 - \cos^2 x = \sin^2 x $$

Apply the reciprocal identity

Using the Reciprocal Identities knowledge point

$$ \sin x = \frac{1}{\csc x} \implies \sin^2 x = \frac{1}{\csc^2 x} $$

Apply the cotangent Pythagorean identity

Using the Pythagorean Identities knowledge point

$$ 1 + \cot^2 x = \csc^2 x \implies \frac{1}{\csc^2 x} = \frac{1}{1 + \cot^2 x} $$

Answer:

Question 1

Algebra

Question 2

Pythagorean

Question 3

Reciprocal

Question 4

Pythagorean