If (secθ + sinθ)/(secθ - sinθ) = (4 + √3)/(4 - √3), 0°< θ < 45°, then, what is the value of (cos2θ + √3tanθ)?

If (secθ + sinθ)/(secθ - sinθ) = (4 + √3)/(4 - √3), 0°< θ < 45°, then, what is the value of (cos2θ + √3tanθ)? Correct Answer 3/2

Given:

(secθ + sinθ)/(secθ - sinθ) = (4 + √3)/(4 - √3), 0°<θ < 45°

Formula Used:

(a +b)/(a - b) = c/d

∴ a/b = (c + d)/(c – d)

Calculation:

(secθ + sinθ)/(secθ - sinθ) = (4 + √3)/(4 - √3)

By componendo and dividendo

⇒ (secθ + sinθ + secθ - sinθ)/(secθ + sinθ - secθ + sinθ) = (4 + √3 + 4 - √3)/(4 + √3 - 4 + √3)

⇒ 2secθ/2sinθ = (2 × 4)/(2 × √3)

⇒ secθ/sinθ = 4/√3

⇒ sinθ/secθ = √3/4

⇒ sinθ cosθ = √3/4

⇒ 2sinθ cosθ = √3/2

⇒ sin2θ = √3/2

⇒ 2θ = 60°

⇒ θ = 30°

The value of (cos2θ + √3tanθ) = cos60° + √3tan30°

⇒ (1/2) + √3(1/√3)

⇒ 1/2 + 1

⇒ 3/2

∴ The correct answer is 3/2