CDS 2025 · PYQ · Trigonometric Identities · medium
What is sin θ/(1 − cot θ) + cos θ/(1 − tan θ) (θ ≠ π/4) equal to?
A.sin θ + cos θ✓ Correct
B.sin θ − cos θ
C.cos θ − sin θ
D.−(sin θ + cos θ)
Explanation
sin θ/(1 − cot θ) = sin θ/(1 − cos θ/sin θ) = sin²θ/(sin θ − cos θ). cos θ/(1 − tan θ) = cos θ/(1 − sin θ/cos θ) = cos²θ/(cos θ − sin θ) = −cos²θ/(sin θ − cos θ). Sum = (sin²θ − cos²θ)/(sin θ − cos θ) = (sin θ − cos θ)(sin θ + cos θ)/(sin θ − cos θ) = sin θ + cos θ.
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