\\frac{1 - \sin\theta}{\cos\theta} + \frac{\cos\theta}{1 - \sin\theta} = 2\sec\theta\

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Identities", "Pythagorean Trigonometric Identity" ], "new_concepts": [], "current_concepts": [ "Trigonometric Identities", "Pythagorean Trigonometric Identity" ] } ## Find a common denominator $$ \frac{1-\sin\theta}{\cos\theta} + \frac{\cos\theta}{1-\sin\theta} = \frac{(1-\sin\theta)^2 + \cos^2\theta}{\cos\theta(1-\sin\theta)} $$ ## Expand and simplify the numerator $$ \begin{aligned} (1-\sin\theta)^2 + \cos^2\theta &= 1 - 2\sin\theta + \sin^2\theta + \cos^2\theta \ &= 1 - 2\sin\theta + 1 \ &= 2 - 2\sin\theta \ &= 2(1-\sin\theta) \end{aligned} $$ ## Reduce the fraction $$ \begin{aligned} \frac{2(1-\sin\theta)}{\cos\theta(1-\sin\theta)} &= \frac{2}{\cos\theta} \ &= 2\sec\theta \end{aligned} $$ To prove the identity: $$ \frac{1-\sin\theta}{\cos\theta} + \frac{\cos\theta}{1-\sin\theta} = 2\sec\theta $$ We simplify the left-hand side (LHS): $$ \begin{aligned} \text{LHS} &= \frac{(1-\sin\theta)^2 + \cos^2\theta}{\cos\theta(1-\sin\theta)} \ &= \frac{1 - 2\sin\theta + \sin^2\theta + \cos^2\theta}{\cos\theta(1-\sin\theta)} \ &= \frac{1 - 2\sin\theta + 1}{\cos\theta(1-\sin\theta)} \ &= \frac{2(1-\sin\theta)}{\cos\theta(1-\sin\theta)} \ &= \frac{2}{\cos\theta} \ &= 2\sec\theta = \text{RHS} \end{aligned} $$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Identities" ] }

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Explanation:

Find a common denominator

Expand and simplify the numerator

Reduce the fraction

Answer: