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Question
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ.
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Solution
L.H.S. = sin θ (1 – tan θ) – cos θ (1 – cot θ)
= `sin θ (1 - (sin θ)/(cos θ)) - cos θ (1 - (cos θ)/(sin θ))`
= `sin θ - (sin^2θ)/(cosθ) - cos θ + (cos^2θ)/(sinθ)`
= `sin θ + (cos^2θ)/(sinθ) - (sin^2θ)/(cosθ) - cos θ`
= `(sin^2θ + cos^2θ)/(sinθ) - ((sin^2θ + cos^2θ)/(cosθ))`
= `1/(sinθ) - 1/(cosθ)` ...[∵ sin2θ + cos2θ = 1]
= cosec θ – sec θ
= R.H.S.
∴ sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ
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