If sin x + cos x = √3 cos x, then the value of cot x is:
1. \(\frac{{\sqrt 3 + 1}}{2}\)
2. √3
3. 1
4. \(\frac{{\sqrt 3 -1}}{2}\)
1. \(\frac{{\sqrt 3 + 1}}{2}\)
2. √3
3. 1
4. \(\frac{{\sqrt 3 -1}}{2}\)
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Correct Answer – Option 1 : \(\frac{{\sqrt 3 + 1}}{2}\)
Given:
sin x + cos x = √3 cos x
Concept used:
Rationalization method used.
Formula used:
Tanθ = Sinθ/Cosθ
a2 – b2 = (a + b) × (a – b)
Calculation:
sin x + cos x = √3 cos x
⇒ (sin x + cos x)/cos x = √3
⇒ (sin x/cos x + cos x/cos x ) = √3
⇒ tan x + 1 = √3
⇒ tan x = √3 – 1
⇒ 1/cot x = √3 – 1
⇒ cot x = 1/(√3 – 1)
⇒ cot x = (√3 + 1)/[(√3 – 1) × (√3 + 1)]
⇒ cot x = (√3 + 1)/[(√3)2 – 1]
⇒ cot x = (√3 + 1)/(3 – 1)
⇒ cot x = (√3 + 1)/2
∴ The value of cot x is (√3 + 1)/2.