If $ cosec \theta = \frac 53 $ , then evaluate ($ sec^2 \theta – 1) \times cot^2 \theta \times (1 + cot^2 \theta)$.

Options
a) $ \frac {9}{16}$
b) $ \frac {25}{9}$
c) $ \frac {25}{16}$
d) $ \frac {4}{5}$

Solution

As per question
Formula :- $ sec^2 \theta – 1 = tan^2 \theta $
$ 1 + cot^2 \theta = cosec^2 \theta $
($ sec^2 \theta – 1) \times cot^2 \theta \times \frac{(1 + cot^2 \theta)}{cosec^2 \theta}$
$ tan^2 \theta \times cot^2 \theta \times cosec^2 \theta $
$ tan^2 \theta \times \frac{1}{tan^2 \theta} \times cosec^2 \theta $
$ cosec^2 \theta $
($ cosec \theta)^2 $
Given = $ cosec \theta = \frac{5}{3}$
$ (\frac {5}{3})^2 = \frac {25}{9}$

Ans is ) $ \frac {25}{9}$