If cosec theta = 5/3 , then evaluate (sec2 theta - 1) x cot2 theta x (1 + cot2 theta).

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 = $cosec \theta = \frac{5}{3}$
$(\frac {5}{3})^2 = \frac {25}{9}$