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Question
If `sin θ + cos θ = sqrt(3)`, then show that tan θ + cot θ = 1.
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Solution
`sin θ + cos θ = sqrt(3)` ...[Given]
∴ (sin θ + cos θ)2 = 3 ...[Squaring on both sides]
∴ sin2θ + 2 sin θ cos θ + cos2θ = 3 ...[∵ (a + b)2 = a2 + 2ab + b2]
∴ (sin2θ + cos2θ) + 2 sin θ cos θ = 3
∴ 1 + 2 sin θ cos θ = 3 ...[∵ sin2θ + cos2θ = 1]
∴ 2 sin θ cos θ = 2
∴ sin θ cos θ = 1 ...(i)
`tan θ + cot θ = (sin θ)/(cos θ) + (cos θ)/(sin θ)`
= `(sin^2θ + cos^2θ)/(cos θ sin θ)`
= `1/(sin θ cos θ)` ...[∵ sin2θ + cos2θ = 1]
= `1/1` ...[From (i)]
= 1
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