Question

prove the identity.

\\((\csc^2 x - 1)\sec x = \cot x \csc x\\)

click here to learn more about writing a trigonometric proof.

statement
\\((\csc^2 x - 1)\sec x\\)

reason
select
algebraic transformation
reciprocal identities
quotient identities
pythagorean identities
odd and even identities

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Trigonometric Proofs",
"Pythagorean Identities"
],
"current_concepts": [
"Trigonometric Proofs",
"Pythagorean Identities",
"Trigonometric Ratios"
]
}
</pre_analysis>

<reasoning>

Apply Pythagorean identity to the expression

\[
\csc^2 x - 1 = \cot^2 x
\]

Substitute and simplify the product

\[
(\csc^2 x - 1)\sec x = \cot^2 x \sec x
\]

Identify the matching reason

The transformation uses the Pythagorean identity \(\cot^2 x + 1 = \csc^2 x\).
</reasoning>

<answer>

• Statement: \(\cot^2 x \sec x\)
• Reason: Pythagorean identities

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Trigonometric Proofs"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Trigonometric Proofs",
"Pythagorean Identities"
],
"current_concepts": [
"Trigonometric Proofs",
"Pythagorean Identities",
"Trigonometric Ratios"
]
}
</pre_analysis>

<reasoning>

Apply Pythagorean identity to the expression

\[
\csc^2 x - 1 = \cot^2 x
\]

Substitute and simplify the product

\[
(\csc^2 x - 1)\sec x = \cot^2 x \sec x
\]

Identify the matching reason

The transformation uses the Pythagorean identity \(\cot^2 x + 1 = \csc^2 x\).
</reasoning>

<answer>

• Statement: \(\cot^2 x \sec x\)
• Reason: Pythagorean identities

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Trigonometric Proofs"
]
}
</post_analysis>