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Question
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ
Solution
L.H.S = sin θ (1 – tan θ) – cos θ (1 – cot θ)
= `sintheta (1 - (sintheta)/(costheta)) - costheta (1 - (costheta)/(sintheta))`
= `sintheta - (sin^2theta)/costheta - costheta + (cos^2theta)/sintheta`
= `sintheta + (cos^2theta)/sintheta - (sin^2theta)/costheta - costheta`
= `(sin^2theta + cos^2theta)/sintheta - ((sin^2theta + cos^2theta)/costheta)`
= `1/sintheta - 1/costheta` ......[∵ 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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