SSC CGL 2021 PYQ — Question 73

The correct answer is option 1 i.e. $\frac{5√3}{3}$

Given :

$\frac{cos^2 \theta}{cot^2\theta + sin^2\theta -1} = 3$

and 0º < θ < 90º.

Concept used :

Trigonometric functions

sin²A + cos²A = 1      ----- (1)

cosec²A - cot²A = 1        ----- (2)

Calculations :

⇒  $\frac{cos^2 \theta}{cot^2\theta + sin^2\theta -1} = 3$

Using equation (1), we get

⇒ $\frac{cos^2\theta}{\frac{cos^2\theta}{sin^2\theta} - cos^2\theta }$ = 3

⇒ $\frac{cos^2\theta}{cos^2\theta(cosec^2\theta - 1)}$ = 3

Using equation (2)

⇒ $\frac{1}{cot^2\theta}$ = 3

⇒ $tan^2\theta = 3$

⇒ $tan \theta = \sqrt3$

⇒ $\theta = tan ^{-1}(\sqrt3)$

⇒ $\theta =$ 60°

Now put the value of $\theta$

⇒  (tanθ + cosecθ)

⇒ tan 60 + cosec 60

⇒ $\sqrt 3 + \frac{2}{\sqrt 3}$

⇒ (3 + 2)/$\sqrt3$

⇒ $5/\sqrt 3$

⇒ $\frac{5\sqrt3}{3}$.