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If 4 - 2sin2θ - 5cosθ = 0, 0°< θ < 90°, then the value of cosθ - tanθ is:  
1. \(\frac{{1 + 2\surd 3}}{2}\)
2. \(\frac{{2 - \surd 3}}{2}\)
3. \(\frac{{2 + \surd 3}}{2}\)
4. \(\frac{{1 - 2\surd 3}}{2}\)

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Correct Answer - Option 4 : \(\frac{{1 - 2\surd 3}}{2}\)

Given:

4 - 2sin2θ - 5cosθ = 0

Formula used:

sin2θ + cos2θ = 1

Calculation:

4 - 2sin2θ - 5cosθ = 0

⇒ 4 - 2(1 - cos2θ ) - 5cosθ = 0

⇒ 4 - 2 - 2cos2θ - 5cosθ = 0

⇒ 2cos2θ - 5cosθ + 2 = 0

⇒ cosθ = [5 ± √(25 - 16)]/4

⇒ cosθ = (5 ± 3)/4

⇒ cosθ = 2 or 1/2

cosθ can't more than 1, cosθ = 1/2

tanθ = √3

cosθ - tanθ

⇒ (1/2) - √3

⇒ \(\frac{{1 - 2\surd 3}}{2}\)

∴ The value is \(\frac{{1 - 2\surd 3}}{2}\).

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