Statement-1:

$cosec^{-1}\frac{3}{2}+cos^{-1}\frac{2}{3}-2cot^{-1}\frac{1}{7}-cot^{-1}7 = cot^{-1}7.$

Statement-2: $sin^{-1} x + cos^{-1} x =\frac{\pi}{2}, tan^{-1} x + cot^{-1} x =\frac{\pi}{2},$

$cosec^{-1}x = sin^{-1}\frac{1}{x}, $ and for $ x > 0, cot^{-1}x = tan^{-1}\frac{1}{x}$

Statement 1 is False, Statement 2 is True.

Statement-2 is true (see theory).

Using statement-2, we have

$cosec^{-1}\frac{3}{2}+cos^{-1}\frac{2}{3}-2cot^{-1}\frac{1}{7}-cot^{-1}7 $

$= \left(sin^{-1}\frac{2}{3}+cos^{-1}\frac{2}{3}\right) - cot^{-1}\frac{1}{7}-(tan^{-1}7 + cot^{-1}7)$

$= \frac{\pi}{2} - cot^{-1}\frac{1}{7}-\frac{\pi}{2} = - cot^{-1}\frac{1}{7}= - tan^{-1}7$

So, statement-1 is not correct.