Question

If secθ= cosec30° and θ is an acute angle, find the value of 4 sin²θ - 2 cos²θ.

Answer 1

secθ= cosec30°
⇒ secθ = 2
⇒ secθ = sec60°
⇒ θ = 60°
Now,
4sin²θ - 2cos²θ
= 4sin²60° - 2cos²60°

= ${\displaystyle 4\times {\left(\frac{\sqrt{3}}{2}\right)}^2-2\times {\left(\frac{1}{2}\right)}^2}$

= ${\displaystyle 4\times \frac{3}{4}-2\times \frac{1}{4}}$

= ${\displaystyle 3-\frac{1}{2}}$

= ${\displaystyle \frac{6-1}{2}}$

= ${\displaystyle \frac{5}{2}}$.