Question

If 8sin²θ + 2cosθ  = 5, 0° < θ < 90°, then the value of tan²θ + sec²θ - sin²θ  will be:
1. \(\frac{153}{72}\)
2. \(\frac{431}{144}\)
3. \(\frac{305}{144}\)
4. \(\frac{23}{9}\)

Answer 1

Correct Answer - Option 3 : \(\frac{305}{144}\)

Formula used:

sin2θ + cos2θ = 1

Calculation:

8sin2θ + 2cosθ  = 5

⇒ 8(1 - cos2θ) + 2cosθ = 5 

⇒ 8 - 8cos2θ + 2cosθ = 5

⇒ 8cos2θ - 2cosθ - 3 = 0

⇒ 8cos2θ - 6cosθ + 4cosθ - 3 =0

⇒ 2cosθ(4cosθ - 3) + 1(4cosθ - 3) = 0

⇒ (2cosθ + 1)(4cosθ - 3) = 0

⇒ cosθ ≠ - 1/2, cosθ = 3/4

∵  cosθ = Base/ Hypotenuse = 3/4 

By pythagoras theorem 

Hypotenue² = Base² + Perpendicular²

⇒ 4² = 3² + P²

⇒ P = \( \sqrt7\)

tan2θ + sec2θ - sin2θ = 7/9 + 16/9 - 7/16

⇒ (368 - 63)/ 144

∴ The required answer = 305/144