COSECANT. 33 



56. Differential coefficient of cosec x. 



Let y = cosec x ; proceed as in the last example, and we 

 find 



dy _ cos x 

 dx sin 2 x ' 



57. Since tana;, cot a;, sec a;, and cosec # are all fractional 

 forms, we may deduce the differential coefficient of each of 

 these functions by Art. 31 from those of sin x and cos x. 

 Thus, let 



sin a; 

 y = tan x = , 



COSiC 



d sin x . d cos x 



, cosec ; sma; j 



,, r ay dx dx . 



therefore -/ = , , Art. 31, 



dx cos x 



cos x + sm^ x 

 cos 2 a; 



1 



, Arts. 51 and 52, 



cos" a; 

 Similarly we may proceed with cot x, sec x, and cosec x. 



Since vers x = 1 cos x, the differential coefficient of vers x 

 by Arts. 27 and 33 



= differential coefficient of cos x 



= sin x. 



T. D. C. 



