( 193 ) 

 DIFFERENTIAL CALCULUS. 



923. Having given 



/7T \ . /27T \ 



sin x sin ( + x ) sin ( +x] ...... 



\m ) \m ) 



sin | (m - 1) + x 1 = 2~ f "~ I ) sin mx ; 

 prove that 



cot x + cot f -f- a; J + cot ( -I- a; J + . . . . 



+ cot I (m - 1) - + a; | = m cot wws; 



924. Prove that the limit of (cos a;) 00 *, as x diminishes in- 

 itely, is e~*' 



Ify = cot- I , 



' ',- (- iyin-1 sinny sin'y. 

 Prove that, if n be a posi: ^cr, the expression 



gf-l 2X*"* 3jC"~" 



(x-n) c'+ 11 * n _ a * la - 3 + 'V + (/l "" 



will be positive for all positive values yf x. 



*&* d l+=, 



then will 



132 



