Given that cos θ = √3/2, then the value of (cosec^2 θ - sec^2 θ)/(cosec^2 θ + sec^2​ ​​​​​​θ) is :

Question

Given that cos θ = √3/2, then the value of \(\frac{cosec^2\,\theta - sec^2\,\theta}{cosec^2\,\theta + sec^2\,\theta}\) is :

(cosec²θ - sec²θ)/(cosec²θ + sec²θ)

(a) -1

(b) 1

(c) 1/2

(d) - 1/2

Answer 1

Correct answer is (c) 1/2

cos θ = √3/2

cos θ = cos 30°

θ = 30°

Now,

\(\frac{cosec^2\,\theta - sec^2\,\theta}{cosec^2\,\theta + sec^2\,\theta}\)

\(=\frac{cosec^2\,30^\circ - sec^2\,\theta}{cosec^2\,30^\circ + sec^2\,\theta}\)

\(=\frac{(2)^2 - \left(\frac{2}{\sqrt{3}}\right)^2}{(2)^2 + \left(\frac{2}{\sqrt{3}}\right)^2}\)

\(=\frac{4 - \frac{4}{3}}{4 + \frac{4}{3}}\)

\(=\cfrac{\frac{12 - 4}{3}}{\frac{12 + 4}{3}}\)

\(=\frac{8}{16}\)

\(\Rightarrow \frac{1}{2}\)